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<div class="title">opennurbs_math.h</div>  </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/* $NoKeywords: $ */</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment">// Copyright (c) 1993-2012 Robert McNeel &amp; Associates. All rights reserved.</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment">// OpenNURBS, Rhinoceros, and Rhino3D are registered trademarks of Robert</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment">// McNeel &amp; Associates.</span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment">// THIS SOFTWARE IS PROVIDED &quot;AS IS&quot; WITHOUT EXPRESS OR IMPLIED WARRANTY.</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment">// ALL IMPLIED WARRANTIES OF FITNESS FOR ANY PARTICULAR PURPOSE AND OF</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment">// MERCHANTABILITY ARE HEREBY DISCLAIMED.</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment">//              </span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment">// For complete openNURBS copyright information see &lt;http://www.opennurbs.org&gt;.</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160; </div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="preprocessor">#if !defined(ON_MATH_INC_)</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="preprocessor">#define ON_MATH_INC_</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160; </div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>;</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n___interval.html">ON_Interval</a>;</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n___line.html">ON_Line</a>;</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n___arc.html">ON_Arc</a>;</div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="keyword">class </span><a class="code" href="class_o_n___plane.html">ON_Plane</a>;</div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160; </div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="comment">  Class for carefully adding long list of numbers.</span></div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00030"></a><span class="lineno"><a class="line" href="class_o_n___sum.html">   30</a></span>&#160;<span class="keyword">class </span>ON_CLASS <a class="code" href="class_o_n___sum.html">ON_Sum</a></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;{</div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;<span class="keyword">public</span>:</div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160; </div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;<span class="comment">    Calls ON_Sum::Begin(x)</span></div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;  <span class="keywordtype">void</span> operator=(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160; </div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment">    Calls ON_Sum::Plus(x);</span></div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;  <span class="keywordtype">void</span> operator+=(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160; </div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;<span class="comment">    Calls ON_Sum::Plus(-x);</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;  <span class="keywordtype">void</span> operator-=(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160; </div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;<span class="comment">    Creates a sum that is ready to be used.</span></div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;  <a class="code" href="class_o_n___sum.html">ON_Sum</a>();</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160; </div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;<span class="comment">    If a sum is being used more than once, call Begin()</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;<span class="comment">    before starting each sum.</span></div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;<span class="comment">    starting_value - [in] Initial value of sum.</span></div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;  <span class="keywordtype">void</span> Begin( <span class="keywordtype">double</span> starting_value = 0.0 );</div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160; </div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;<span class="comment">    Add x to the current sum.</span></div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;<span class="comment">    x - [in] value to add to the current sum.</span></div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;  <span class="keywordtype">void</span> Plus( <span class="keywordtype">double</span> x );</div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160; </div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;<span class="comment">    Calculates the total sum.   </span></div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;<span class="comment">    error_estimate - [out] if not NULL, the returned value of</span></div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;<span class="comment">       *error_estimate is an estimate of the error in the sum.</span></div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;<span class="comment">    Total of the sum.</span></div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;<span class="comment">  Remarks:</span></div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;<span class="comment">    You can get subtotals by mixing calls to Plus() and Total().</span></div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;<span class="comment">    In delicate sums, some precision may be lost in the final</span></div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;<span class="comment">    total if you call Total() to calculate subtotals.</span></div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;  <span class="keywordtype">double</span> Total( <span class="keywordtype">double</span>* error_estimate = NULL );</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160; </div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;<span class="comment">    Number of summands.</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  <span class="keywordtype">int</span> SummandCount() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160; </div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;<span class="keyword">private</span>:</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;    sum1_max_count=256,</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;    sum2_max_count=512,</div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;    sum3_max_count=1024</div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;  };</div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;  <span class="keywordtype">double</span> m_sum_err;</div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;  <span class="keywordtype">double</span> m_pos_sum;     </div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;  <span class="keywordtype">double</span> m_neg_sum;  </div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;  </div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;  <span class="keywordtype">int</span> m_zero_count; <span class="comment">// number of zeros added</span></div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;  <span class="keywordtype">int</span> m_pos_count; <span class="comment">// number of positive numbers added</span></div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;  <span class="keywordtype">int</span> m_neg_count; <span class="comment">// number of negative numbers added</span></div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;  </div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;  <span class="keywordtype">int</span> m_pos_sum1_count;</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;  <span class="keywordtype">int</span> m_pos_sum2_count;</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;  <span class="keywordtype">int</span> m_pos_sum3_count;</div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;  <span class="keywordtype">double</span> m_pos_sum1[sum1_max_count];</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;  <span class="keywordtype">double</span> m_pos_sum2[sum2_max_count];</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;  <span class="keywordtype">double</span> m_pos_sum3[sum3_max_count];</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;  </div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;  <span class="keywordtype">int</span> m_neg_sum1_count;</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;  <span class="keywordtype">int</span> m_neg_sum2_count;</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;  <span class="keywordtype">int</span> m_neg_sum3_count;</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;  <span class="keywordtype">double</span> m_neg_sum1[sum1_max_count];</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;  <span class="keywordtype">double</span> m_neg_sum2[sum2_max_count];</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;  <span class="keywordtype">double</span> m_neg_sum3[sum3_max_count];</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160; </div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;  <span class="keywordtype">double</span> SortAndSum( <span class="keywordtype">int</span>, <span class="keywordtype">double</span>* );</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;};</div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160; </div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;<span class="comment">  Abstract function with an arbitrary number of parameters</span></div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;<span class="comment">  and values.  ON_Evaluator is used to pass functions to</span></div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;<span class="comment">  local solvers.</span></div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00133"></a><span class="lineno"><a class="line" href="class_o_n___evaluator.html">  133</a></span>&#160;<span class="keyword">class </span>ON_CLASS <a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a></div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;{</div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="keyword">public</span>:</div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160; </div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;<span class="comment">    Construction of the class for a function that takes</span></div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;<span class="comment">    parameter_count input functions and returns</span></div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;<span class="comment">    value_count values.  If the domain is infinite, pass</span></div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;<span class="comment">    a NULL for the domain[] and periodic[] arrays.  If</span></div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;<span class="comment">    the domain is finite, pass a domain[] array with</span></div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;<span class="comment">    parameter_count increasing intervals.  If one or more of</span></div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;<span class="comment">    the parameters is periodic, pass the fundamental domain</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;<span class="comment">    in the domain[] array and a true in the periodic[] array.</span></div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;<span class="comment">    parameter_count - [in] &gt;= 1.  Number of input parameters</span></div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;<span class="comment">    value_count - [in] &gt;= 1.  Number of output values.</span></div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;<span class="comment">    domain - [in] If not NULL, then this is an array</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;<span class="comment">                  of parameter_count increasing intervals</span></div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;<span class="comment">                  that defines the domain of the function.</span></div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;<span class="comment">    periodic - [in] if not NULL, then this is an array of </span></div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;<span class="comment">                parameter_count bools where b[i] is true if</span></div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;<span class="comment">                the i-th parameter is periodic.  Valid </span></div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;<span class="comment">                increasing finite domains must be specificed</span></div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;<span class="comment">                when this parameter is not NULL.</span></div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;  <a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a>( </div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;    <span class="keywordtype">int</span> parameter_count,</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;    <span class="keywordtype">int</span> value_count,</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n___interval.html">ON_Interval</a>* domain,</div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">bool</span>* periodic</div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;    );</div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160; </div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;  <span class="keyword">virtual</span> ~<a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a>();</div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;  </div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;<span class="comment">    Evaluate the function that takes m_parameter_count parameters</span></div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;<span class="comment">    and returns a m_value_count dimensional point.</span></div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;<span class="comment">    parameters - [in] array of m_parameter_count evaluation parameters</span></div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;<span class="comment">    values - [out] array of m_value_count function values</span></div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;<span class="comment">    jacobian - [out] If NULL, simply evaluate the value of the function.</span></div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;<span class="comment">                     If not NULL, this is the jacobian of the function.</span></div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;<span class="comment">                     jacobian[i][j] = j-th partial of the i-th value</span></div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;<span class="comment">                     0 &lt;= i &lt; m_value_count,</span></div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;<span class="comment">                     0 &lt;= j &lt; m_parameter_count</span></div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;<span class="comment">                     If not NULL, then all the memory for the</span></div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;<span class="comment">                     jacobian is allocated, you just need to fill</span></div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;<span class="comment">                     in the answers.</span></div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;<span class="comment">  Example:</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;<span class="comment">    If f(u,v) = square of the distance from a fixed point P to a </span></div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;<span class="comment">    surface evaluated at (u,v), then</span></div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;<span class="comment">          values[0] = (S-P)o(S-P)</span></div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;<span class="comment">          jacobian[0] = ( 2*(Du o (S-P)), 2*(Dv o (S-P)) )</span></div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;<span class="comment">    where S, Du, Dv = surface point and first partials evaluated</span></div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;<span class="comment">    at u=parameters[0], v = parameters[1].</span></div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;<span class="comment">    If the function takes 3 parameters, say (x,y,z), and returns</span></div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;<span class="comment">    two values, say f(x,y,z) and g(z,y,z), then</span></div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;<span class="comment">          values[0] = f(x,y,z)</span></div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;<span class="comment">          values[1] = g(x,y,z)</span></div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;<span class="comment">          jacobian[0] = (DfDx, DfDy, DfDz)</span></div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;<span class="comment">          jacobian[1] = (DgDx, DgDy, DgDz)</span></div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;<span class="comment">    where dfx denotes the first partial of f with respect to x.</span></div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;<span class="comment">    0 = unable to evaluate</span></div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;<span class="comment">    1 = successful evaluation</span></div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;<span class="comment">    2 = found answer, terminate search</span></div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;  <span class="keyword">virtual</span> <span class="keywordtype">int</span> Evaluate(</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;       <span class="keyword">const</span> <span class="keywordtype">double</span>* parameters,</div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;       <span class="keywordtype">double</span>* values,</div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;       <span class="keywordtype">double</span>** jacobian</div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;       ) = 0;</div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160; </div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;<span class="comment">    OPTIONAL ability to evaluate the hessian in the case when </span></div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;<span class="comment">    m_value_count is one.  If your function has more that</span></div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;<span class="comment">    one value or it is not feasable to evaluate the hessian,</span></div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;<span class="comment">    then do not override this function.  The default implementation</span></div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;<span class="comment">    returns -1.</span></div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;<span class="comment">  Parameters:</span></div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;<span class="comment">    parameters - [in] array of m_parameter_count evaluation parameters</span></div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;<span class="comment">    value - [out] value of the function (one double)</span></div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;<span class="comment">    gradient - [out] The gradient of the function.  This is a vector</span></div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;<span class="comment">                     of length m_parameter_count; gradient[i] is</span></div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;<span class="comment">                     the first partial of the function with respect to</span></div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;<span class="comment">                     the i-th parameter.</span></div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;<span class="comment">    hessian - [out] The hessian of the function. This is an</span></div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;<span class="comment">                    m_parameter_count x m_parameter_count </span></div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;<span class="comment">                    symmetric matrix: hessian[i][j] is the</span></div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;<span class="comment">                    second partial of the function with respect</span></div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;<span class="comment">                    to the i-th and j-th parameters.  The evaluator</span></div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;<span class="comment">                    is responsible for filling in both the upper</span></div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;<span class="comment">                    and lower triangles.  Since the matrix is</span></div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;<span class="comment">                    symmetrix, you should do something like evaluate</span></div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;<span class="comment">                    the upper triangle and copy the values to the</span></div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;<span class="comment">                    lower tiangle.</span></div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;<span class="comment">   -1 = Hessian evaluation not available.</span></div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;<span class="comment">    0 = unable to evaluate</span></div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;<span class="comment">    1 = successful evaluation</span></div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;<span class="comment">    2 = found answer, terminate search</span></div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;  <span class="keyword">virtual</span> <span class="keywordtype">int</span> EvaluateHessian(</div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;       <span class="keyword">const</span> <span class="keywordtype">double</span>* parameters,</div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;       <span class="keywordtype">double</span>* value,</div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;       <span class="keywordtype">double</span>* gradient,</div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;       <span class="keywordtype">double</span>** hessian</div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;       );</div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;  </div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;  <span class="comment">// Number of the function&#39;s input parameters. This number</span></div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;  <span class="comment">// is &gt;= 1 and is specified in the constructor.</span></div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">int</span> m_parameter_count;</div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160; </div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;  <span class="comment">// Number of the function&#39;s output values. This number</span></div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;  <span class="comment">// is &gt;= 1 and is specified in the constructor.</span></div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;  <span class="keyword">const</span> <span class="keywordtype">int</span> m_value_count;</div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160; </div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;<span class="comment">    Functions can have finite or infinite domains. Finite domains</span></div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;<span class="comment">    are specified by passing the domain[] array to the constructor</span></div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;<span class="comment">    or filling in the m_domain[] member variable.  If</span></div>
<div class="line"><a name="l00265"></a><span class="lineno">  265</span>&#160;<span class="comment">    m_domain.Count() == m_parameter_count &gt; 0, then the function</span></div>
<div class="line"><a name="l00266"></a><span class="lineno">  266</span>&#160;<span class="comment">    has finite domains.</span></div>
<div class="line"><a name="l00267"></a><span class="lineno">  267</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00268"></a><span class="lineno">  268</span>&#160;<span class="comment">    True if the domain of the function is finite.</span></div>
<div class="line"><a name="l00269"></a><span class="lineno">  269</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00270"></a><span class="lineno">  270</span>&#160;  <span class="keywordtype">bool</span> FiniteDomain() <span class="keyword">const</span>;</div>
<div class="line"><a name="l00271"></a><span class="lineno">  271</span>&#160; </div>
<div class="line"><a name="l00272"></a><span class="lineno">  272</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00273"></a><span class="lineno">  273</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00274"></a><span class="lineno">  274</span>&#160;<span class="comment">    If a function has a periodic parameter, then the m_domain</span></div>
<div class="line"><a name="l00275"></a><span class="lineno">  275</span>&#160;<span class="comment">    interval for that parameter is the fundamental domain and</span></div>
<div class="line"><a name="l00276"></a><span class="lineno">  276</span>&#160;<span class="comment">    the m_bPeriodicParameter bool for that parameter is true.</span></div>
<div class="line"><a name="l00277"></a><span class="lineno">  277</span>&#160;<span class="comment">    A parameter is periodic if, and only if, </span></div>
<div class="line"><a name="l00278"></a><span class="lineno">  278</span>&#160;<span class="comment">    m_domain.Count() == m_parameter_count, and </span></div>
<div class="line"><a name="l00279"></a><span class="lineno">  279</span>&#160;<span class="comment">    m_bPeriodicParameter.Count() == m_parameter_count, and</span></div>
<div class="line"><a name="l00280"></a><span class="lineno">  280</span>&#160;<span class="comment">    m_bPeriodicParameter[parameter_index] is true.</span></div>
<div class="line"><a name="l00281"></a><span class="lineno">  281</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00282"></a><span class="lineno">  282</span>&#160;<span class="comment">    True if the function parameter is periodic.</span></div>
<div class="line"><a name="l00283"></a><span class="lineno">  283</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00284"></a><span class="lineno">  284</span>&#160;  <span class="keywordtype">bool</span> Periodic(</div>
<div class="line"><a name="l00285"></a><span class="lineno">  285</span>&#160;    <span class="keywordtype">int</span> parameter_index</div>
<div class="line"><a name="l00286"></a><span class="lineno">  286</span>&#160;    ) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00287"></a><span class="lineno">  287</span>&#160; </div>
<div class="line"><a name="l00288"></a><span class="lineno">  288</span>&#160;  <span class="comment">/*</span></div>
<div class="line"><a name="l00289"></a><span class="lineno">  289</span>&#160;<span class="comment">  Description:</span></div>
<div class="line"><a name="l00290"></a><span class="lineno">  290</span>&#160;<span class="comment">    If a function has a periodic parameter, then the m_domain</span></div>
<div class="line"><a name="l00291"></a><span class="lineno">  291</span>&#160;<span class="comment">    interval for that parameter is the fundamental domain and</span></div>
<div class="line"><a name="l00292"></a><span class="lineno">  292</span>&#160;<span class="comment">    the m_bPeriodicParameter bool for that parameter is true.</span></div>
<div class="line"><a name="l00293"></a><span class="lineno">  293</span>&#160;<span class="comment">    A parameter is periodic if, and only if, </span></div>
<div class="line"><a name="l00294"></a><span class="lineno">  294</span>&#160;<span class="comment">    m_domain.Count() == m_parameter_count, and </span></div>
<div class="line"><a name="l00295"></a><span class="lineno">  295</span>&#160;<span class="comment">    m_bPeriodicParameter.Count() == m_parameter_count, and</span></div>
<div class="line"><a name="l00296"></a><span class="lineno">  296</span>&#160;<span class="comment">    m_bPeriodicParameter[parameter_index] is true.</span></div>
<div class="line"><a name="l00297"></a><span class="lineno">  297</span>&#160;<span class="comment">  Returns:</span></div>
<div class="line"><a name="l00298"></a><span class="lineno">  298</span>&#160;<span class="comment">    The domain of the parameter.  If the domain is infinite,</span></div>
<div class="line"><a name="l00299"></a><span class="lineno">  299</span>&#160;<span class="comment">    the (-1.0e300, +1.0e300) is returned.</span></div>
<div class="line"><a name="l00300"></a><span class="lineno">  300</span>&#160;<span class="comment">  */</span></div>
<div class="line"><a name="l00301"></a><span class="lineno">  301</span>&#160;  <a class="code" href="class_o_n___interval.html">ON_Interval</a> Domain(</div>
<div class="line"><a name="l00302"></a><span class="lineno">  302</span>&#160;    <span class="keywordtype">int</span> parameter_index</div>
<div class="line"><a name="l00303"></a><span class="lineno">  303</span>&#160;    ) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00304"></a><span class="lineno">  304</span>&#160; </div>
<div class="line"><a name="l00305"></a><span class="lineno">  305</span>&#160; </div>
<div class="line"><a name="l00306"></a><span class="lineno">  306</span>&#160;  <span class="comment">// If the function has a finite domain or periodic</span></div>
<div class="line"><a name="l00307"></a><span class="lineno">  307</span>&#160;  <span class="comment">// parameters, then m_domain[] is an array of </span></div>
<div class="line"><a name="l00308"></a><span class="lineno">  308</span>&#160;  <span class="comment">// m_parameter_count finite increasing intervals.</span></div>
<div class="line"><a name="l00309"></a><span class="lineno">  309</span>&#160;  <a class="code" href="class_o_n___simple_array.html">ON_SimpleArray&lt;ON_Interval&gt;</a> m_domain;</div>
<div class="line"><a name="l00310"></a><span class="lineno">  310</span>&#160; </div>
<div class="line"><a name="l00311"></a><span class="lineno">  311</span>&#160;  <span class="comment">// If the function has periodic parameters, then </span></div>
<div class="line"><a name="l00312"></a><span class="lineno">  312</span>&#160;  <span class="comment">// m_bPeriodicParameter[] is an array of m_parameter_count</span></div>
<div class="line"><a name="l00313"></a><span class="lineno">  313</span>&#160;  <span class="comment">// bools.  If m_bPeriodicParameter[i] is true, then</span></div>
<div class="line"><a name="l00314"></a><span class="lineno">  314</span>&#160;  <span class="comment">// the i-th parameter is periodic and m_domain[i] is </span></div>
<div class="line"><a name="l00315"></a><span class="lineno">  315</span>&#160;  <span class="comment">// the fundamental domain for that parameter.</span></div>
<div class="line"><a name="l00316"></a><span class="lineno">  316</span>&#160;  <a class="code" href="class_o_n___simple_array.html">ON_SimpleArray&lt;bool&gt;</a> m_bPeriodicParameter;</div>
<div class="line"><a name="l00317"></a><span class="lineno">  317</span>&#160; </div>
<div class="line"><a name="l00318"></a><span class="lineno">  318</span>&#160;<span class="keyword">private</span>:</div>
<div class="line"><a name="l00319"></a><span class="lineno">  319</span>&#160;  <a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a>(); <span class="comment">// prohibit default constructor</span></div>
<div class="line"><a name="l00320"></a><span class="lineno">  320</span>&#160;  <a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a>&amp; operator=(<span class="keyword">const</span> <a class="code" href="class_o_n___evaluator.html">ON_Evaluator</a>&amp;); <span class="comment">// prohibit operator= (can&#39;t copy const members)</span></div>
<div class="line"><a name="l00321"></a><span class="lineno">  321</span>&#160;};</div>
<div class="line"><a name="l00322"></a><span class="lineno">  322</span>&#160; </div>
<div class="line"><a name="l00323"></a><span class="lineno">  323</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00324"></a><span class="lineno">  324</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00325"></a><span class="lineno">  325</span>&#160;<span class="comment">  Test a double to make sure it is a valid number.</span></div>
<div class="line"><a name="l00326"></a><span class="lineno">  326</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00327"></a><span class="lineno">  327</span>&#160;<span class="comment">  True if x != ON_UNSET_VALUE and _finite(x) is true.</span></div>
<div class="line"><a name="l00328"></a><span class="lineno">  328</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00329"></a><span class="lineno">  329</span>&#160;ON_DECL</div>
<div class="line"><a name="l00330"></a><span class="lineno">  330</span>&#160;<span class="keywordtype">bool</span> ON_IsValid( <span class="keywordtype">double</span> x );</div>
<div class="line"><a name="l00331"></a><span class="lineno">  331</span>&#160; </div>
<div class="line"><a name="l00332"></a><span class="lineno">  332</span>&#160;ON_DECL</div>
<div class="line"><a name="l00333"></a><span class="lineno">  333</span>&#160;<span class="keywordtype">bool</span> ON_IsValidFloat( <span class="keywordtype">float</span> x );</div>
<div class="line"><a name="l00334"></a><span class="lineno">  334</span>&#160; </div>
<div class="line"><a name="l00335"></a><span class="lineno">  335</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00336"></a><span class="lineno">  336</span>&#160;<span class="comment">class ON_CLASS ON_TimeLimit</span></div>
<div class="line"><a name="l00337"></a><span class="lineno">  337</span>&#160;<span class="comment">{</span></div>
<div class="line"><a name="l00338"></a><span class="lineno">  338</span>&#160;<span class="comment">  ON_TimeLimit();</span></div>
<div class="line"><a name="l00339"></a><span class="lineno">  339</span>&#160;<span class="comment">  ON_TimeLimit(ON__UINT64 time_limit_seconds);</span></div>
<div class="line"><a name="l00340"></a><span class="lineno">  340</span>&#160;<span class="comment">  void SetTimeLimit(ON__UINT64 time_limit_seconds);</span></div>
<div class="line"><a name="l00341"></a><span class="lineno">  341</span>&#160;<span class="comment">  bool Continue() const;</span></div>
<div class="line"><a name="l00342"></a><span class="lineno">  342</span>&#160;<span class="comment">  bool IsSet() const;</span></div>
<div class="line"><a name="l00343"></a><span class="lineno">  343</span>&#160;<span class="comment">private:</span></div>
<div class="line"><a name="l00344"></a><span class="lineno">  344</span>&#160;<span class="comment">  ON__UINT64 m_time_limit[2];</span></div>
<div class="line"><a name="l00345"></a><span class="lineno">  345</span>&#160;<span class="comment">};</span></div>
<div class="line"><a name="l00346"></a><span class="lineno">  346</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00347"></a><span class="lineno">  347</span>&#160; </div>
<div class="line"><a name="l00348"></a><span class="lineno">  348</span>&#160;<span class="comment">// The ON_IS_FINITE and ON_IS_VALID defines are much faster</span></div>
<div class="line"><a name="l00349"></a><span class="lineno">  349</span>&#160;<span class="comment">// than calling ON_IsValid(), but need to be used when</span></div>
<div class="line"><a name="l00350"></a><span class="lineno">  350</span>&#160;<span class="comment">// the macro expansion works.</span></div>
<div class="line"><a name="l00351"></a><span class="lineno">  351</span>&#160; </div>
<div class="line"><a name="l00352"></a><span class="lineno">  352</span>&#160;<span class="preprocessor">#if   defined(ON_LITTLE_ENDIAN)</span></div>
<div class="line"><a name="l00353"></a><span class="lineno">  353</span>&#160; </div>
<div class="line"><a name="l00354"></a><span class="lineno">  354</span>&#160;<span class="comment">// works on little endian CPUs with IEEE doubles</span></div>
<div class="line"><a name="l00355"></a><span class="lineno">  355</span>&#160;<span class="preprocessor">#define ON_IS_FINITE(x) (0x7FF0 != (*((unsigned short*)(&amp;x) + 3) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00356"></a><span class="lineno">  356</span>&#160;<span class="preprocessor">#define ON_IS_VALID(x)  (x != ON_UNSET_VALUE &amp;&amp; 0x7FF0 != (*((unsigned short*)(&amp;x) + 3) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00357"></a><span class="lineno">  357</span>&#160;<span class="preprocessor">#define ON_IS_VALID_FLOAT(x)  (x != ON_UNSET_FLOAT)</span></div>
<div class="line"><a name="l00358"></a><span class="lineno">  358</span>&#160;<span class="comment">//TODO - ADD FAST ugly bit check#define ON_IS_VALID_FLOAT(x)  (x != ON_UNSET_FLOAT &amp;&amp; 0x7FF0 != (*((unsigned short*)(&amp;x) + 3) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00359"></a><span class="lineno">  359</span>&#160; </div>
<div class="line"><a name="l00360"></a><span class="lineno">  360</span>&#160;<span class="preprocessor">#elif defined(ON_BIG_ENDIAN)</span></div>
<div class="line"><a name="l00361"></a><span class="lineno">  361</span>&#160; </div>
<div class="line"><a name="l00362"></a><span class="lineno">  362</span>&#160;<span class="comment">// works on big endian CPUs with IEEE doubles</span></div>
<div class="line"><a name="l00363"></a><span class="lineno">  363</span>&#160;<span class="preprocessor">#define ON_IS_FINITE(x) (0x7FF0 != (*((unsigned short*)(&amp;x)) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00364"></a><span class="lineno">  364</span>&#160;<span class="preprocessor">#define ON_IS_VALID(x)  (x != ON_UNSET_VALUE &amp;&amp; 0x7FF0 != (*((unsigned short*)(&amp;x)) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00365"></a><span class="lineno">  365</span>&#160;<span class="preprocessor">#define ON_IS_VALID_FLOAT(x)  (x != ON_UNSET_FLOAT)</span></div>
<div class="line"><a name="l00366"></a><span class="lineno">  366</span>&#160;<span class="comment">//TODO - ADD FAST ugly bit check#define ON_IS_VALID_FLOAT(x)  (x != ON_UNSET_FLOAT &amp;&amp; 0x7FF0 != (*((unsigned short*)(&amp;x) + 3) &amp; 0x7FF0))</span></div>
<div class="line"><a name="l00367"></a><span class="lineno">  367</span>&#160; </div>
<div class="line"><a name="l00368"></a><span class="lineno">  368</span>&#160;<span class="preprocessor">#else</span></div>
<div class="line"><a name="l00369"></a><span class="lineno">  369</span>&#160; </div>
<div class="line"><a name="l00370"></a><span class="lineno">  370</span>&#160;<span class="comment">// Returns true if x is a finite double.  Specifically,</span></div>
<div class="line"><a name="l00371"></a><span class="lineno">  371</span>&#160;<span class="comment">// _finite returns a nonzero value (true) if its argument x</span></div>
<div class="line"><a name="l00372"></a><span class="lineno">  372</span>&#160;<span class="comment">// is not infinite, that is, if -INF &lt; x &lt; +INF. </span></div>
<div class="line"><a name="l00373"></a><span class="lineno">  373</span>&#160;<span class="comment">// It returns 0 (false) if the argument is infinite or a NaN.</span></div>
<div class="line"><a name="l00374"></a><span class="lineno">  374</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00375"></a><span class="lineno">  375</span>&#160;<span class="comment">// If you are trying to compile opennurbs on a platform</span></div>
<div class="line"><a name="l00376"></a><span class="lineno">  376</span>&#160;<span class="comment">// that does not support finite(), then see if you can</span></div>
<div class="line"><a name="l00377"></a><span class="lineno">  377</span>&#160;<span class="comment">// use _fpclass(), fpclass(), _isnan(), or isnan().  If</span></div>
<div class="line"><a name="l00378"></a><span class="lineno">  378</span>&#160;<span class="comment">// you can&#39;t find anything, then just set this</span></div>
<div class="line"><a name="l00379"></a><span class="lineno">  379</span>&#160;<span class="comment">// function to return true.</span></div>
<div class="line"><a name="l00380"></a><span class="lineno">  380</span>&#160; </div>
<div class="line"><a name="l00381"></a><span class="lineno">  381</span>&#160;<span class="preprocessor">#if defined(_GNU_SOURCE)</span></div>
<div class="line"><a name="l00382"></a><span class="lineno">  382</span>&#160;<span class="comment">// if you are using an older version of gcc, use finite()</span></div>
<div class="line"><a name="l00383"></a><span class="lineno">  383</span>&#160;<span class="comment">//#define ON_IS_FINITE(x) (finite(x)?true:false)</span></div>
<div class="line"><a name="l00384"></a><span class="lineno">  384</span>&#160;<span class="preprocessor">#define ON_IS_FINITE(x) (isfinite(x)?true:false)</span></div>
<div class="line"><a name="l00385"></a><span class="lineno">  385</span>&#160;<span class="preprocessor">#else</span></div>
<div class="line"><a name="l00386"></a><span class="lineno">  386</span>&#160;<span class="preprocessor">#define ON_IS_FINITE(x) (_finite(x)?true:false)</span></div>
<div class="line"><a name="l00387"></a><span class="lineno">  387</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00388"></a><span class="lineno">  388</span>&#160; </div>
<div class="line"><a name="l00389"></a><span class="lineno">  389</span>&#160;<span class="preprocessor">#define ON_IS_VALID(x)  (x != ON_UNSET_VALUE &amp;&amp; ON_IS_FINITE(x))</span></div>
<div class="line"><a name="l00390"></a><span class="lineno">  390</span>&#160;<span class="preprocessor">#define ON_IS_VALID_FLOAT(x)  (x != ON_UNSET_FLOAT &amp;&amp; ON_IS_FINITE(x))</span></div>
<div class="line"><a name="l00391"></a><span class="lineno">  391</span>&#160; </div>
<div class="line"><a name="l00392"></a><span class="lineno">  392</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00393"></a><span class="lineno">  393</span>&#160; </div>
<div class="line"><a name="l00394"></a><span class="lineno">  394</span>&#160; </div>
<div class="line"><a name="l00395"></a><span class="lineno">  395</span>&#160;ON_DECL</div>
<div class="line"><a name="l00396"></a><span class="lineno">  396</span>&#160;<span class="keywordtype">float</span> ON_ArrayDotProduct( <span class="comment">// returns AoB</span></div>
<div class="line"><a name="l00397"></a><span class="lineno">  397</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00398"></a><span class="lineno">  398</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">float</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00399"></a><span class="lineno">  399</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">float</span>*  <span class="comment">// B[]</span></div>
<div class="line"><a name="l00400"></a><span class="lineno">  400</span>&#160;          );</div>
<div class="line"><a name="l00401"></a><span class="lineno">  401</span>&#160; </div>
<div class="line"><a name="l00402"></a><span class="lineno">  402</span>&#160;ON_DECL</div>
<div class="line"><a name="l00403"></a><span class="lineno">  403</span>&#160;<span class="keywordtype">void</span>   ON_ArrayScale( </div>
<div class="line"><a name="l00404"></a><span class="lineno">  404</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00405"></a><span class="lineno">  405</span>&#160;          <span class="keywordtype">float</span>,        <span class="comment">// a</span></div>
<div class="line"><a name="l00406"></a><span class="lineno">  406</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">float</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00407"></a><span class="lineno">  407</span>&#160;          <span class="keywordtype">float</span>*        <span class="comment">// returns a*A[]</span></div>
<div class="line"><a name="l00408"></a><span class="lineno">  408</span>&#160;          );</div>
<div class="line"><a name="l00409"></a><span class="lineno">  409</span>&#160; </div>
<div class="line"><a name="l00410"></a><span class="lineno">  410</span>&#160;ON_DECL</div>
<div class="line"><a name="l00411"></a><span class="lineno">  411</span>&#160;<span class="keywordtype">void</span>   ON_Array_aA_plus_B( </div>
<div class="line"><a name="l00412"></a><span class="lineno">  412</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00413"></a><span class="lineno">  413</span>&#160;          <span class="keywordtype">float</span>,        <span class="comment">// a</span></div>
<div class="line"><a name="l00414"></a><span class="lineno">  414</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">float</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00415"></a><span class="lineno">  415</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">float</span>*, <span class="comment">// B[]</span></div>
<div class="line"><a name="l00416"></a><span class="lineno">  416</span>&#160;          <span class="keywordtype">float</span>*        <span class="comment">// returns a*A[] + B[]</span></div>
<div class="line"><a name="l00417"></a><span class="lineno">  417</span>&#160;          );</div>
<div class="line"><a name="l00418"></a><span class="lineno">  418</span>&#160; </div>
<div class="line"><a name="l00419"></a><span class="lineno">  419</span>&#160;ON_DECL</div>
<div class="line"><a name="l00420"></a><span class="lineno">  420</span>&#160;<span class="keywordtype">double</span> ON_ArrayDotProduct( <span class="comment">// returns AoB</span></div>
<div class="line"><a name="l00421"></a><span class="lineno">  421</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00422"></a><span class="lineno">  422</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00423"></a><span class="lineno">  423</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// B[]</span></div>
<div class="line"><a name="l00424"></a><span class="lineno">  424</span>&#160;          );</div>
<div class="line"><a name="l00425"></a><span class="lineno">  425</span>&#160; </div>
<div class="line"><a name="l00426"></a><span class="lineno">  426</span>&#160;ON_DECL</div>
<div class="line"><a name="l00427"></a><span class="lineno">  427</span>&#160;<span class="keywordtype">double</span> ON_ArrayDotDifference( <span class="comment">// returns A o ( B - C )</span></div>
<div class="line"><a name="l00428"></a><span class="lineno">  428</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00429"></a><span class="lineno">  429</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00430"></a><span class="lineno">  430</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// B[]</span></div>
<div class="line"><a name="l00431"></a><span class="lineno">  431</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// C[]</span></div>
<div class="line"><a name="l00432"></a><span class="lineno">  432</span>&#160;          );</div>
<div class="line"><a name="l00433"></a><span class="lineno">  433</span>&#160; </div>
<div class="line"><a name="l00434"></a><span class="lineno">  434</span>&#160;ON_DECL</div>
<div class="line"><a name="l00435"></a><span class="lineno">  435</span>&#160;<span class="keywordtype">double</span> ON_ArrayMagnitude( <span class="comment">// returns sqrt(AoA)</span></div>
<div class="line"><a name="l00436"></a><span class="lineno">  436</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00437"></a><span class="lineno">  437</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// A[]</span></div>
<div class="line"><a name="l00438"></a><span class="lineno">  438</span>&#160;          );</div>
<div class="line"><a name="l00439"></a><span class="lineno">  439</span>&#160; </div>
<div class="line"><a name="l00440"></a><span class="lineno">  440</span>&#160;ON_DECL</div>
<div class="line"><a name="l00441"></a><span class="lineno">  441</span>&#160;<span class="keywordtype">double</span> ON_ArrayMagnitudeSquared( <span class="comment">// returns AoA</span></div>
<div class="line"><a name="l00442"></a><span class="lineno">  442</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00443"></a><span class="lineno">  443</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// A[]</span></div>
<div class="line"><a name="l00444"></a><span class="lineno">  444</span>&#160;          );</div>
<div class="line"><a name="l00445"></a><span class="lineno">  445</span>&#160; </div>
<div class="line"><a name="l00446"></a><span class="lineno">  446</span>&#160;ON_DECL</div>
<div class="line"><a name="l00447"></a><span class="lineno">  447</span>&#160;<span class="keywordtype">double</span> ON_ArrayDistance( <span class="comment">// returns sqrt((A-B)o(A-B))</span></div>
<div class="line"><a name="l00448"></a><span class="lineno">  448</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00449"></a><span class="lineno">  449</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00450"></a><span class="lineno">  450</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// B[]</span></div>
<div class="line"><a name="l00451"></a><span class="lineno">  451</span>&#160;          );</div>
<div class="line"><a name="l00452"></a><span class="lineno">  452</span>&#160; </div>
<div class="line"><a name="l00453"></a><span class="lineno">  453</span>&#160;ON_DECL</div>
<div class="line"><a name="l00454"></a><span class="lineno">  454</span>&#160;<span class="keywordtype">double</span> ON_ArrayDistanceSquared( <span class="comment">// returns (A-B)o(A-B)</span></div>
<div class="line"><a name="l00455"></a><span class="lineno">  455</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00456"></a><span class="lineno">  456</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00457"></a><span class="lineno">  457</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// B[]</span></div>
<div class="line"><a name="l00458"></a><span class="lineno">  458</span>&#160;          );</div>
<div class="line"><a name="l00459"></a><span class="lineno">  459</span>&#160; </div>
<div class="line"><a name="l00460"></a><span class="lineno">  460</span>&#160;ON_DECL</div>
<div class="line"><a name="l00461"></a><span class="lineno">  461</span>&#160;<span class="keywordtype">void</span>   ON_ArrayScale( </div>
<div class="line"><a name="l00462"></a><span class="lineno">  462</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00463"></a><span class="lineno">  463</span>&#160;          <span class="keywordtype">double</span>,        <span class="comment">// a</span></div>
<div class="line"><a name="l00464"></a><span class="lineno">  464</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00465"></a><span class="lineno">  465</span>&#160;          <span class="keywordtype">double</span>*        <span class="comment">// returns a*A[]</span></div>
<div class="line"><a name="l00466"></a><span class="lineno">  466</span>&#160;          );</div>
<div class="line"><a name="l00467"></a><span class="lineno">  467</span>&#160; </div>
<div class="line"><a name="l00468"></a><span class="lineno">  468</span>&#160;ON_DECL</div>
<div class="line"><a name="l00469"></a><span class="lineno">  469</span>&#160;<span class="keywordtype">void</span>   ON_Array_aA_plus_B( </div>
<div class="line"><a name="l00470"></a><span class="lineno">  470</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// size of arrays (can be zero)</span></div>
<div class="line"><a name="l00471"></a><span class="lineno">  471</span>&#160;          <span class="keywordtype">double</span>,        <span class="comment">// a</span></div>
<div class="line"><a name="l00472"></a><span class="lineno">  472</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// A[]</span></div>
<div class="line"><a name="l00473"></a><span class="lineno">  473</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// B[]</span></div>
<div class="line"><a name="l00474"></a><span class="lineno">  474</span>&#160;          <span class="keywordtype">double</span>*        <span class="comment">// returns a*A[] + B[]</span></div>
<div class="line"><a name="l00475"></a><span class="lineno">  475</span>&#160;          );</div>
<div class="line"><a name="l00476"></a><span class="lineno">  476</span>&#160; </div>
<div class="line"><a name="l00477"></a><span class="lineno">  477</span>&#160;ON_DECL</div>
<div class="line"><a name="l00478"></a><span class="lineno">  478</span>&#160;<span class="keywordtype">int</span>    ON_SearchMonotoneArray( <span class="comment">// find a value in an increasing array</span></div>
<div class="line"><a name="l00479"></a><span class="lineno">  479</span>&#160;          <span class="comment">// returns  -1: t &lt; array[0]</span></div>
<div class="line"><a name="l00480"></a><span class="lineno">  480</span>&#160;          <span class="comment">//           i: array[i] &lt;= t &lt; array[i+1] ( 0 &lt;= i &lt; length-1 )</span></div>
<div class="line"><a name="l00481"></a><span class="lineno">  481</span>&#160;          <span class="comment">//    length-1: t == array[length-1]</span></div>
<div class="line"><a name="l00482"></a><span class="lineno">  482</span>&#160;          <span class="comment">//      length: t &gt;= array[length-1]</span></div>
<div class="line"><a name="l00483"></a><span class="lineno">  483</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// array[]</span></div>
<div class="line"><a name="l00484"></a><span class="lineno">  484</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// length of array</span></div>
<div class="line"><a name="l00485"></a><span class="lineno">  485</span>&#160;          <span class="keywordtype">double</span>         <span class="comment">// t = value to search for</span></div>
<div class="line"><a name="l00486"></a><span class="lineno">  486</span>&#160;          );</div>
<div class="line"><a name="l00487"></a><span class="lineno">  487</span>&#160; </div>
<div class="line"><a name="l00488"></a><span class="lineno">  488</span>&#160; </div>
<div class="line"><a name="l00489"></a><span class="lineno">  489</span>&#160;<span class="comment">/* </span></div>
<div class="line"><a name="l00490"></a><span class="lineno">  490</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00491"></a><span class="lineno">  491</span>&#160;<span class="comment">  Compute a binomial coefficient.</span></div>
<div class="line"><a name="l00492"></a><span class="lineno">  492</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00493"></a><span class="lineno">  493</span>&#160;<span class="comment">  i - [in]</span></div>
<div class="line"><a name="l00494"></a><span class="lineno">  494</span>&#160;<span class="comment">  j - [in]</span></div>
<div class="line"><a name="l00495"></a><span class="lineno">  495</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00496"></a><span class="lineno">  496</span>&#160;<span class="comment">  (i+j)!/(i!j!), if 0 &lt;= i and 0 &lt;= j, and 0 otherwise.</span></div>
<div class="line"><a name="l00497"></a><span class="lineno">  497</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l00498"></a><span class="lineno">  498</span>&#160;<span class="comment">  ON_TrinomialCoefficient()</span></div>
<div class="line"><a name="l00499"></a><span class="lineno">  499</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l00500"></a><span class="lineno">  500</span>&#160;<span class="comment">  If (i+j) &lt;= 52, this function is fast and returns the exact</span></div>
<div class="line"><a name="l00501"></a><span class="lineno">  501</span>&#160;<span class="comment">  value of the binomial coefficient.  </span></div>
<div class="line"><a name="l00502"></a><span class="lineno">  502</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00503"></a><span class="lineno">  503</span>&#160;<span class="comment">  For (i+j) &gt; 52, the coefficient is computed recursively using</span></div>
<div class="line"><a name="l00504"></a><span class="lineno">  504</span>&#160;<span class="comment">  the formula  bc(i,j) = bc(i-1,j) + bc(i,j-1).</span></div>
<div class="line"><a name="l00505"></a><span class="lineno">  505</span>&#160;<span class="comment">  For (i+j) much larger than 60, this is inefficient.</span></div>
<div class="line"><a name="l00506"></a><span class="lineno">  506</span>&#160;<span class="comment">  If you need binomial coefficients for large i and j, then you</span></div>
<div class="line"><a name="l00507"></a><span class="lineno">  507</span>&#160;<span class="comment">  should probably be using something like Stirling&#39;s Formula.  </span></div>
<div class="line"><a name="l00508"></a><span class="lineno">  508</span>&#160;<span class="comment">  (Look up &quot;Stirling&quot; or &quot;Gamma function&quot; in a calculus book.)</span></div>
<div class="line"><a name="l00509"></a><span class="lineno">  509</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00510"></a><span class="lineno">  510</span>&#160;ON_DECL</div>
<div class="line"><a name="l00511"></a><span class="lineno">  511</span>&#160;<span class="keywordtype">double</span> ON_BinomialCoefficient( </div>
<div class="line"><a name="l00512"></a><span class="lineno">  512</span>&#160;          <span class="keywordtype">int</span> i,</div>
<div class="line"><a name="l00513"></a><span class="lineno">  513</span>&#160;          <span class="keywordtype">int</span> j</div>
<div class="line"><a name="l00514"></a><span class="lineno">  514</span>&#160;          );</div>
<div class="line"><a name="l00515"></a><span class="lineno">  515</span>&#160; </div>
<div class="line"><a name="l00516"></a><span class="lineno">  516</span>&#160; </div>
<div class="line"><a name="l00517"></a><span class="lineno">  517</span>&#160;<span class="comment">/* </span></div>
<div class="line"><a name="l00518"></a><span class="lineno">  518</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00519"></a><span class="lineno">  519</span>&#160;<span class="comment">  Compute a trinomial coefficient.</span></div>
<div class="line"><a name="l00520"></a><span class="lineno">  520</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00521"></a><span class="lineno">  521</span>&#160;<span class="comment">  i - [in]</span></div>
<div class="line"><a name="l00522"></a><span class="lineno">  522</span>&#160;<span class="comment">  j - [in]</span></div>
<div class="line"><a name="l00523"></a><span class="lineno">  523</span>&#160;<span class="comment">  k - [in]</span></div>
<div class="line"><a name="l00524"></a><span class="lineno">  524</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00525"></a><span class="lineno">  525</span>&#160;<span class="comment">  (i+j+k)!/(i!j!k!), if 0 &lt;= i, 0 &lt;= j and 0&lt;= k, and 0 otherwise.</span></div>
<div class="line"><a name="l00526"></a><span class="lineno">  526</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l00527"></a><span class="lineno">  527</span>&#160;<span class="comment">  ON_BinomialCoefficient()</span></div>
<div class="line"><a name="l00528"></a><span class="lineno">  528</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l00529"></a><span class="lineno">  529</span>&#160;<span class="comment">  The trinomial coefficient is computed using the formula</span></div>
<div class="line"><a name="l00530"></a><span class="lineno">  530</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00531"></a><span class="lineno">  531</span>&#160;<span class="comment">          (i+j+k)!      (i+j+k)!       (j+k)!</span></div>
<div class="line"><a name="l00532"></a><span class="lineno">  532</span>&#160;<span class="comment">          --------   =  --------   *  -------</span></div>
<div class="line"><a name="l00533"></a><span class="lineno">  533</span>&#160;<span class="comment">          i! j! k!      i! (j+k)!      j! k!</span></div>
<div class="line"><a name="l00534"></a><span class="lineno">  534</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00535"></a><span class="lineno">  535</span>&#160;<span class="comment">                      = ON_BinomialCoefficient(i,j+k)*ON_BinomialCoefficient(j,k)</span></div>
<div class="line"><a name="l00536"></a><span class="lineno">  536</span>&#160;<span class="comment">  </span></div>
<div class="line"><a name="l00537"></a><span class="lineno">  537</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00538"></a><span class="lineno">  538</span>&#160;ON_DECL</div>
<div class="line"><a name="l00539"></a><span class="lineno">  539</span>&#160;<span class="keywordtype">double</span> ON_TrinomialCoefficient( </div>
<div class="line"><a name="l00540"></a><span class="lineno">  540</span>&#160;          <span class="keywordtype">int</span> i,</div>
<div class="line"><a name="l00541"></a><span class="lineno">  541</span>&#160;          <span class="keywordtype">int</span> j,</div>
<div class="line"><a name="l00542"></a><span class="lineno">  542</span>&#160;          <span class="keywordtype">int</span> k</div>
<div class="line"><a name="l00543"></a><span class="lineno">  543</span>&#160;          );</div>
<div class="line"><a name="l00544"></a><span class="lineno">  544</span>&#160; </div>
<div class="line"><a name="l00545"></a><span class="lineno">  545</span>&#160; </div>
<div class="line"><a name="l00546"></a><span class="lineno">  546</span>&#160;ON_DECL</div>
<div class="line"><a name="l00547"></a><span class="lineno">  547</span>&#160;ON_BOOL32 ON_GetParameterTolerance(</div>
<div class="line"><a name="l00548"></a><span class="lineno">  548</span>&#160;        <span class="keywordtype">double</span>, <span class="keywordtype">double</span>, <span class="comment">// domain</span></div>
<div class="line"><a name="l00549"></a><span class="lineno">  549</span>&#160;        <span class="keywordtype">double</span>,          <span class="comment">// parameter in domain</span></div>
<div class="line"><a name="l00550"></a><span class="lineno">  550</span>&#160;        <span class="keywordtype">double</span>*, <span class="keywordtype">double</span>* <span class="comment">// parameter tolerance (tminus, tplus) returned here</span></div>
<div class="line"><a name="l00551"></a><span class="lineno">  551</span>&#160;        );</div>
<div class="line"><a name="l00552"></a><span class="lineno">  552</span>&#160; </div>
<div class="line"><a name="l00553"></a><span class="lineno">  553</span>&#160; </div>
<div class="line"><a name="l00554"></a><span class="lineno">  554</span>&#160;ON_DECL</div>
<div class="line"><a name="l00555"></a><span class="lineno">  555</span>&#160;ON_BOOL32 ON_IsValidPointList(</div>
<div class="line"><a name="l00556"></a><span class="lineno">  556</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00557"></a><span class="lineno">  557</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00558"></a><span class="lineno">  558</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00559"></a><span class="lineno">  559</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00560"></a><span class="lineno">  560</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">float</span>*</div>
<div class="line"><a name="l00561"></a><span class="lineno">  561</span>&#160;        );</div>
<div class="line"><a name="l00562"></a><span class="lineno">  562</span>&#160; </div>
<div class="line"><a name="l00563"></a><span class="lineno">  563</span>&#160;ON_DECL</div>
<div class="line"><a name="l00564"></a><span class="lineno">  564</span>&#160;ON_BOOL32 ON_IsValidPointList(</div>
<div class="line"><a name="l00565"></a><span class="lineno">  565</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00566"></a><span class="lineno">  566</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00567"></a><span class="lineno">  567</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00568"></a><span class="lineno">  568</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00569"></a><span class="lineno">  569</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*</div>
<div class="line"><a name="l00570"></a><span class="lineno">  570</span>&#160;        );</div>
<div class="line"><a name="l00571"></a><span class="lineno">  571</span>&#160; </div>
<div class="line"><a name="l00572"></a><span class="lineno">  572</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00573"></a><span class="lineno">  573</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00574"></a><span class="lineno">  574</span>&#160;<span class="comment">  Determine if a list of points is planar.</span></div>
<div class="line"><a name="l00575"></a><span class="lineno">  575</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00576"></a><span class="lineno">  576</span>&#160;<span class="comment">  bRational - [in]</span></div>
<div class="line"><a name="l00577"></a><span class="lineno">  577</span>&#160;<span class="comment">    false if the points are euclidean (x,y,z)</span></div>
<div class="line"><a name="l00578"></a><span class="lineno">  578</span>&#160;<span class="comment">    true if the points are homogeneous rational (x,y,z,w)</span></div>
<div class="line"><a name="l00579"></a><span class="lineno">  579</span>&#160;<span class="comment">  point_count - [in]</span></div>
<div class="line"><a name="l00580"></a><span class="lineno">  580</span>&#160;<span class="comment">    number of points</span></div>
<div class="line"><a name="l00581"></a><span class="lineno">  581</span>&#160;<span class="comment">  point_stride - [in]</span></div>
<div class="line"><a name="l00582"></a><span class="lineno">  582</span>&#160;<span class="comment">    number of doubles between point x coordinates</span></div>
<div class="line"><a name="l00583"></a><span class="lineno">  583</span>&#160;<span class="comment">    first point&#39;s x coordinate = points[0],</span></div>
<div class="line"><a name="l00584"></a><span class="lineno">  584</span>&#160;<span class="comment">    second point&#39;s x coordinate = points[point_stride],...</span></div>
<div class="line"><a name="l00585"></a><span class="lineno">  585</span>&#160;<span class="comment">  points - [in]</span></div>
<div class="line"><a name="l00586"></a><span class="lineno">  586</span>&#160;<span class="comment">    point coordinates (3d or 4d homogeneous rational)</span></div>
<div class="line"><a name="l00587"></a><span class="lineno">  587</span>&#160;<span class="comment">  boxMin - [in]</span></div>
<div class="line"><a name="l00588"></a><span class="lineno">  588</span>&#160;<span class="comment">  boxMax - [in]</span></div>
<div class="line"><a name="l00589"></a><span class="lineno">  589</span>&#160;<span class="comment">    optional 3d bounding box - pass nulls if not readily available</span></div>
<div class="line"><a name="l00590"></a><span class="lineno">  590</span>&#160;<span class="comment">  tolerance - [in] &gt;= 0.0</span></div>
<div class="line"><a name="l00591"></a><span class="lineno">  591</span>&#160;<span class="comment">  plane_equation0 - [in]</span></div>
<div class="line"><a name="l00592"></a><span class="lineno">  592</span>&#160;<span class="comment">    If you want to test for planarity in a specific plane,</span></div>
<div class="line"><a name="l00593"></a><span class="lineno">  593</span>&#160;<span class="comment">    pass the plane equation in here.  If you want to find</span></div>
<div class="line"><a name="l00594"></a><span class="lineno">  594</span>&#160;<span class="comment">    a plane containing the points, pass null here.</span></div>
<div class="line"><a name="l00595"></a><span class="lineno">  595</span>&#160;<span class="comment">  plane_equation - [out]</span></div>
<div class="line"><a name="l00596"></a><span class="lineno">  596</span>&#160;<span class="comment">    If this point is not null, then the equation of the plane</span></div>
<div class="line"><a name="l00597"></a><span class="lineno">  597</span>&#160;<span class="comment">    containing the points is retuened here.</span></div>
<div class="line"><a name="l00598"></a><span class="lineno">  598</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00599"></a><span class="lineno">  599</span>&#160;<span class="comment">  0 - points are not coplanar to the specified tolerance</span></div>
<div class="line"><a name="l00600"></a><span class="lineno">  600</span>&#160;<span class="comment">  1 - points are coplanar to the specified tolerance</span></div>
<div class="line"><a name="l00601"></a><span class="lineno">  601</span>&#160;<span class="comment">  2 - points are colinear to the specified tolerance</span></div>
<div class="line"><a name="l00602"></a><span class="lineno">  602</span>&#160;<span class="comment">      (in this case, plane_equation is not a unique answer)</span></div>
<div class="line"><a name="l00603"></a><span class="lineno">  603</span>&#160;<span class="comment">  3 - points are coincident to the specified tolerance</span></div>
<div class="line"><a name="l00604"></a><span class="lineno">  604</span>&#160;<span class="comment">      (in this case, plane_equation is not a unique answer)</span></div>
<div class="line"><a name="l00605"></a><span class="lineno">  605</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00606"></a><span class="lineno">  606</span>&#160;ON_DECL</div>
<div class="line"><a name="l00607"></a><span class="lineno">  607</span>&#160;<span class="keywordtype">int</span> ON_IsPointListPlanar(</div>
<div class="line"><a name="l00608"></a><span class="lineno">  608</span>&#160;    <span class="keywordtype">bool</span> bRational,</div>
<div class="line"><a name="l00609"></a><span class="lineno">  609</span>&#160;    <span class="keywordtype">int</span> count,</div>
<div class="line"><a name="l00610"></a><span class="lineno">  610</span>&#160;    <span class="keywordtype">int</span> stride,</div>
<div class="line"><a name="l00611"></a><span class="lineno">  611</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* points,</div>
<div class="line"><a name="l00612"></a><span class="lineno">  612</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* boxMin,</div>
<div class="line"><a name="l00613"></a><span class="lineno">  613</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* boxMax,</div>
<div class="line"><a name="l00614"></a><span class="lineno">  614</span>&#160;    <span class="keywordtype">double</span> tolerance,</div>
<div class="line"><a name="l00615"></a><span class="lineno">  615</span>&#160;    <a class="code" href="class_o_n___plane_equation.html">ON_PlaneEquation</a>* plane_equation</div>
<div class="line"><a name="l00616"></a><span class="lineno">  616</span>&#160;    );</div>
<div class="line"><a name="l00617"></a><span class="lineno">  617</span>&#160; </div>
<div class="line"><a name="l00618"></a><span class="lineno">  618</span>&#160;ON_DECL</div>
<div class="line"><a name="l00619"></a><span class="lineno">  619</span>&#160;ON_BOOL32 ON_IsValidPointGrid(</div>
<div class="line"><a name="l00620"></a><span class="lineno">  620</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00621"></a><span class="lineno">  621</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00622"></a><span class="lineno">  622</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_count0, point_count1,</span></div>
<div class="line"><a name="l00623"></a><span class="lineno">  623</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_stride0, point_stride1,</span></div>
<div class="line"><a name="l00624"></a><span class="lineno">  624</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*</div>
<div class="line"><a name="l00625"></a><span class="lineno">  625</span>&#160;        );</div>
<div class="line"><a name="l00626"></a><span class="lineno">  626</span>&#160; </div>
<div class="line"><a name="l00627"></a><span class="lineno">  627</span>&#160;ON_DECL</div>
<div class="line"><a name="l00628"></a><span class="lineno">  628</span>&#160;<span class="keywordtype">bool</span> ON_ReversePointList(</div>
<div class="line"><a name="l00629"></a><span class="lineno">  629</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00630"></a><span class="lineno">  630</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00631"></a><span class="lineno">  631</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00632"></a><span class="lineno">  632</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00633"></a><span class="lineno">  633</span>&#160;        <span class="keywordtype">double</span>*</div>
<div class="line"><a name="l00634"></a><span class="lineno">  634</span>&#160;        );</div>
<div class="line"><a name="l00635"></a><span class="lineno">  635</span>&#160; </div>
<div class="line"><a name="l00636"></a><span class="lineno">  636</span>&#160;ON_DECL</div>
<div class="line"><a name="l00637"></a><span class="lineno">  637</span>&#160;ON_BOOL32 ON_ReversePointGrid(</div>
<div class="line"><a name="l00638"></a><span class="lineno">  638</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00639"></a><span class="lineno">  639</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00640"></a><span class="lineno">  640</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_count0, point_count1,</span></div>
<div class="line"><a name="l00641"></a><span class="lineno">  641</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_stride0, point_stride1,</span></div>
<div class="line"><a name="l00642"></a><span class="lineno">  642</span>&#160;        <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00643"></a><span class="lineno">  643</span>&#160;        <span class="keywordtype">int</span>       <span class="comment">// dir = 0 or 1</span></div>
<div class="line"><a name="l00644"></a><span class="lineno">  644</span>&#160;        );</div>
<div class="line"><a name="l00645"></a><span class="lineno">  645</span>&#160; </div>
<div class="line"><a name="l00646"></a><span class="lineno">  646</span>&#160;ON_DECL</div>
<div class="line"><a name="l00647"></a><span class="lineno">  647</span>&#160;<span class="keywordtype">bool</span> ON_SwapPointListCoordinates( </div>
<div class="line"><a name="l00648"></a><span class="lineno">  648</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// count</span></div>
<div class="line"><a name="l00649"></a><span class="lineno">  649</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// stride</span></div>
<div class="line"><a name="l00650"></a><span class="lineno">  650</span>&#160;        <span class="keywordtype">float</span>*,</div>
<div class="line"><a name="l00651"></a><span class="lineno">  651</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span> <span class="comment">// coordinates to swap</span></div>
<div class="line"><a name="l00652"></a><span class="lineno">  652</span>&#160;        );</div>
<div class="line"><a name="l00653"></a><span class="lineno">  653</span>&#160; </div>
<div class="line"><a name="l00654"></a><span class="lineno">  654</span>&#160;ON_DECL</div>
<div class="line"><a name="l00655"></a><span class="lineno">  655</span>&#160;<span class="keywordtype">bool</span> ON_SwapPointListCoordinates( </div>
<div class="line"><a name="l00656"></a><span class="lineno">  656</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// count</span></div>
<div class="line"><a name="l00657"></a><span class="lineno">  657</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// stride</span></div>
<div class="line"><a name="l00658"></a><span class="lineno">  658</span>&#160;        <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00659"></a><span class="lineno">  659</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span> <span class="comment">// coordinates to swap</span></div>
<div class="line"><a name="l00660"></a><span class="lineno">  660</span>&#160;        );</div>
<div class="line"><a name="l00661"></a><span class="lineno">  661</span>&#160; </div>
<div class="line"><a name="l00662"></a><span class="lineno">  662</span>&#160;ON_DECL</div>
<div class="line"><a name="l00663"></a><span class="lineno">  663</span>&#160;ON_BOOL32 ON_SwapPointGridCoordinates(</div>
<div class="line"><a name="l00664"></a><span class="lineno">  664</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_count0, point_count1,</span></div>
<div class="line"><a name="l00665"></a><span class="lineno">  665</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_stride0, point_stride1,</span></div>
<div class="line"><a name="l00666"></a><span class="lineno">  666</span>&#160;        <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00667"></a><span class="lineno">  667</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span> <span class="comment">// coordinates to swap</span></div>
<div class="line"><a name="l00668"></a><span class="lineno">  668</span>&#160;        );</div>
<div class="line"><a name="l00669"></a><span class="lineno">  669</span>&#160; </div>
<div class="line"><a name="l00670"></a><span class="lineno">  670</span>&#160;ON_DECL</div>
<div class="line"><a name="l00671"></a><span class="lineno">  671</span>&#160;<span class="keywordtype">bool</span> ON_TransformPointList(</div>
<div class="line"><a name="l00672"></a><span class="lineno">  672</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00673"></a><span class="lineno">  673</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00674"></a><span class="lineno">  674</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00675"></a><span class="lineno">  675</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00676"></a><span class="lineno">  676</span>&#160;        <span class="keywordtype">float</span>*,</div>
<div class="line"><a name="l00677"></a><span class="lineno">  677</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;</div>
<div class="line"><a name="l00678"></a><span class="lineno">  678</span>&#160;        );</div>
<div class="line"><a name="l00679"></a><span class="lineno">  679</span>&#160; </div>
<div class="line"><a name="l00680"></a><span class="lineno">  680</span>&#160;ON_DECL</div>
<div class="line"><a name="l00681"></a><span class="lineno">  681</span>&#160;<span class="keywordtype">bool</span> ON_TransformPointList(</div>
<div class="line"><a name="l00682"></a><span class="lineno">  682</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00683"></a><span class="lineno">  683</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00684"></a><span class="lineno">  684</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00685"></a><span class="lineno">  685</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00686"></a><span class="lineno">  686</span>&#160;        <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00687"></a><span class="lineno">  687</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;</div>
<div class="line"><a name="l00688"></a><span class="lineno">  688</span>&#160;        );</div>
<div class="line"><a name="l00689"></a><span class="lineno">  689</span>&#160; </div>
<div class="line"><a name="l00690"></a><span class="lineno">  690</span>&#160;ON_DECL</div>
<div class="line"><a name="l00691"></a><span class="lineno">  691</span>&#160;ON_BOOL32 ON_TransformPointGrid(</div>
<div class="line"><a name="l00692"></a><span class="lineno">  692</span>&#160;        <span class="keywordtype">int</span>,      <span class="comment">// dim</span></div>
<div class="line"><a name="l00693"></a><span class="lineno">  693</span>&#160;        ON_BOOL32,     <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00694"></a><span class="lineno">  694</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_count0, point_count1,</span></div>
<div class="line"><a name="l00695"></a><span class="lineno">  695</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_stride0, point_stride1,</span></div>
<div class="line"><a name="l00696"></a><span class="lineno">  696</span>&#160;        <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00697"></a><span class="lineno">  697</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;</div>
<div class="line"><a name="l00698"></a><span class="lineno">  698</span>&#160;        );</div>
<div class="line"><a name="l00699"></a><span class="lineno">  699</span>&#160; </div>
<div class="line"><a name="l00700"></a><span class="lineno">  700</span>&#160;ON_DECL</div>
<div class="line"><a name="l00701"></a><span class="lineno">  701</span>&#160;ON_BOOL32 ON_TransformVectorList(</div>
<div class="line"><a name="l00702"></a><span class="lineno">  702</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00703"></a><span class="lineno">  703</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00704"></a><span class="lineno">  704</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00705"></a><span class="lineno">  705</span>&#160;       <span class="keywordtype">float</span>*,</div>
<div class="line"><a name="l00706"></a><span class="lineno">  706</span>&#160;       <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;</div>
<div class="line"><a name="l00707"></a><span class="lineno">  707</span>&#160;       );</div>
<div class="line"><a name="l00708"></a><span class="lineno">  708</span>&#160; </div>
<div class="line"><a name="l00709"></a><span class="lineno">  709</span>&#160;ON_DECL</div>
<div class="line"><a name="l00710"></a><span class="lineno">  710</span>&#160;ON_BOOL32 ON_TransformVectorList(</div>
<div class="line"><a name="l00711"></a><span class="lineno">  711</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00712"></a><span class="lineno">  712</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00713"></a><span class="lineno">  713</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00714"></a><span class="lineno">  714</span>&#160;       <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00715"></a><span class="lineno">  715</span>&#160;       <span class="keyword">const</span> <a class="code" href="class_o_n___xform.html">ON_Xform</a>&amp;</div>
<div class="line"><a name="l00716"></a><span class="lineno">  716</span>&#160;       );</div>
<div class="line"><a name="l00717"></a><span class="lineno">  717</span>&#160; </div>
<div class="line"><a name="l00718"></a><span class="lineno">  718</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00719"></a><span class="lineno">  719</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00720"></a><span class="lineno">  720</span>&#160;<span class="comment">  dim - [in]</span></div>
<div class="line"><a name="l00721"></a><span class="lineno">  721</span>&#160;<span class="comment">    &gt;= 1</span></div>
<div class="line"><a name="l00722"></a><span class="lineno">  722</span>&#160;<span class="comment">  is_rat - [in]</span></div>
<div class="line"><a name="l00723"></a><span class="lineno">  723</span>&#160;<span class="comment">    true if the points are rational and points[dim] is the &quot;weight&quot;</span></div>
<div class="line"><a name="l00724"></a><span class="lineno">  724</span>&#160;<span class="comment">  pointA - [in]</span></div>
<div class="line"><a name="l00725"></a><span class="lineno">  725</span>&#160;<span class="comment">  pointB - [in]</span></div>
<div class="line"><a name="l00726"></a><span class="lineno">  726</span>&#160;<span class="comment">    point coordinates</span></div>
<div class="line"><a name="l00727"></a><span class="lineno">  727</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00728"></a><span class="lineno">  728</span>&#160;<span class="comment">  True if the input is valid and for each coordinate pair,</span></div>
<div class="line"><a name="l00729"></a><span class="lineno">  729</span>&#160;<span class="comment">  |a-b| &lt;= ON_ZERO_TOLERANCE </span></div>
<div class="line"><a name="l00730"></a><span class="lineno">  730</span>&#160;<span class="comment">  or |a-b| &lt;= (fabs(a)+fabs(b))*ON_RELATIVE_TOLERANCE.</span></div>
<div class="line"><a name="l00731"></a><span class="lineno">  731</span>&#160;<span class="comment">  False otherwise.</span></div>
<div class="line"><a name="l00732"></a><span class="lineno">  732</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00733"></a><span class="lineno">  733</span>&#160;ON_DECL</div>
<div class="line"><a name="l00734"></a><span class="lineno">  734</span>&#160;<span class="keywordtype">bool</span> ON_PointsAreCoincident(</div>
<div class="line"><a name="l00735"></a><span class="lineno">  735</span>&#160;    <span class="keywordtype">int</span> dim,</div>
<div class="line"><a name="l00736"></a><span class="lineno">  736</span>&#160;    <span class="keywordtype">int</span> is_rat,</div>
<div class="line"><a name="l00737"></a><span class="lineno">  737</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* pointA,</div>
<div class="line"><a name="l00738"></a><span class="lineno">  738</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* pointB</div>
<div class="line"><a name="l00739"></a><span class="lineno">  739</span>&#160;    );</div>
<div class="line"><a name="l00740"></a><span class="lineno">  740</span>&#160; </div>
<div class="line"><a name="l00741"></a><span class="lineno">  741</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l00742"></a><span class="lineno">  742</span>&#160;<span class="comment">Description</span></div>
<div class="line"><a name="l00743"></a><span class="lineno">  743</span>&#160;<span class="comment">  See ON_PointsAreCoincident() for a description of when opennurbs</span></div>
<div class="line"><a name="l00744"></a><span class="lineno">  744</span>&#160;<span class="comment">  considers two points to be conincident.</span></div>
<div class="line"><a name="l00745"></a><span class="lineno">  745</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00746"></a><span class="lineno">  746</span>&#160;<span class="comment">  dim - [in]</span></div>
<div class="line"><a name="l00747"></a><span class="lineno">  747</span>&#160;<span class="comment">    &gt;= 1</span></div>
<div class="line"><a name="l00748"></a><span class="lineno">  748</span>&#160;<span class="comment">  is_rat - [in]</span></div>
<div class="line"><a name="l00749"></a><span class="lineno">  749</span>&#160;<span class="comment">    true if the points are rational and points[dim] is the &quot;weight&quot;</span></div>
<div class="line"><a name="l00750"></a><span class="lineno">  750</span>&#160;<span class="comment">  point_count - [in]</span></div>
<div class="line"><a name="l00751"></a><span class="lineno">  751</span>&#160;<span class="comment">    number of points &gt;= 2</span></div>
<div class="line"><a name="l00752"></a><span class="lineno">  752</span>&#160;<span class="comment">  point_stride - [in]</span></div>
<div class="line"><a name="l00753"></a><span class="lineno">  753</span>&#160;<span class="comment">    &gt;= (0 != is_rat) ? (dim+1) : dim</span></div>
<div class="line"><a name="l00754"></a><span class="lineno">  754</span>&#160;<span class="comment">  points - [in]</span></div>
<div class="line"><a name="l00755"></a><span class="lineno">  755</span>&#160;<span class="comment">    point coordinates</span></div>
<div class="line"><a name="l00756"></a><span class="lineno">  756</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00757"></a><span class="lineno">  757</span>&#160;<span class="comment">  True if the first and last points are coincident and all other</span></div>
<div class="line"><a name="l00758"></a><span class="lineno">  758</span>&#160;<span class="comment">  points in the list are coincident with the previous point.</span></div>
<div class="line"><a name="l00759"></a><span class="lineno">  759</span>&#160;<span class="comment">  False if there are points that are not coincident or</span></div>
<div class="line"><a name="l00760"></a><span class="lineno">  760</span>&#160;<span class="comment">  point_count &lt; 2 or other input parameters are invalid.</span></div>
<div class="line"><a name="l00761"></a><span class="lineno">  761</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00762"></a><span class="lineno">  762</span>&#160;ON_DECL</div>
<div class="line"><a name="l00763"></a><span class="lineno">  763</span>&#160;<span class="keywordtype">bool</span> ON_PointsAreCoincident(</div>
<div class="line"><a name="l00764"></a><span class="lineno">  764</span>&#160;    <span class="keywordtype">int</span> dim,</div>
<div class="line"><a name="l00765"></a><span class="lineno">  765</span>&#160;    <span class="keywordtype">int</span> is_rat,</div>
<div class="line"><a name="l00766"></a><span class="lineno">  766</span>&#160;    <span class="keywordtype">int</span> point_count,</div>
<div class="line"><a name="l00767"></a><span class="lineno">  767</span>&#160;    <span class="keywordtype">int</span> point_stride,</div>
<div class="line"><a name="l00768"></a><span class="lineno">  768</span>&#160;    <span class="keyword">const</span> <span class="keywordtype">double</span>* points</div>
<div class="line"><a name="l00769"></a><span class="lineno">  769</span>&#160;    );</div>
<div class="line"><a name="l00770"></a><span class="lineno">  770</span>&#160; </div>
<div class="line"><a name="l00771"></a><span class="lineno">  771</span>&#160;ON_DECL</div>
<div class="line"><a name="l00772"></a><span class="lineno">  772</span>&#160;<span class="keywordtype">int</span> ON_ComparePoint( <span class="comment">// returns </span></div>
<div class="line"><a name="l00773"></a><span class="lineno">  773</span>&#160;                              <span class="comment">// -1: first &lt; second</span></div>
<div class="line"><a name="l00774"></a><span class="lineno">  774</span>&#160;                              <span class="comment">//  0: first == second</span></div>
<div class="line"><a name="l00775"></a><span class="lineno">  775</span>&#160;                              <span class="comment">// +1: first &gt; second</span></div>
<div class="line"><a name="l00776"></a><span class="lineno">  776</span>&#160;          <span class="keywordtype">int</span> dim,            <span class="comment">// dim (&gt;=0)</span></div>
<div class="line"><a name="l00777"></a><span class="lineno">  777</span>&#160;          ON_BOOL32 israt,    <span class="comment">// true for rational CVs</span></div>
<div class="line"><a name="l00778"></a><span class="lineno">  778</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>* cv0,  <span class="comment">// first CV</span></div>
<div class="line"><a name="l00779"></a><span class="lineno">  779</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>* cv1   <span class="comment">// secont CV</span></div>
<div class="line"><a name="l00780"></a><span class="lineno">  780</span>&#160;          );</div>
<div class="line"><a name="l00781"></a><span class="lineno">  781</span>&#160; </div>
<div class="line"><a name="l00782"></a><span class="lineno">  782</span>&#160;ON_DECL</div>
<div class="line"><a name="l00783"></a><span class="lineno">  783</span>&#160;<span class="keywordtype">int</span> ON_ComparePointList( <span class="comment">// returns </span></div>
<div class="line"><a name="l00784"></a><span class="lineno">  784</span>&#160;                              <span class="comment">// -1: first &lt; second</span></div>
<div class="line"><a name="l00785"></a><span class="lineno">  785</span>&#160;                              <span class="comment">//  0: first == second</span></div>
<div class="line"><a name="l00786"></a><span class="lineno">  786</span>&#160;                              <span class="comment">// +1: first &gt; second</span></div>
<div class="line"><a name="l00787"></a><span class="lineno">  787</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// dim (&gt;=0)</span></div>
<div class="line"><a name="l00788"></a><span class="lineno">  788</span>&#160;          ON_BOOL32,          <span class="comment">// true for rational CVs</span></div>
<div class="line"><a name="l00789"></a><span class="lineno">  789</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// count</span></div>
<div class="line"><a name="l00790"></a><span class="lineno">  790</span>&#160;          <span class="comment">// first point list</span></div>
<div class="line"><a name="l00791"></a><span class="lineno">  791</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// stride</span></div>
<div class="line"><a name="l00792"></a><span class="lineno">  792</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// point</span></div>
<div class="line"><a name="l00793"></a><span class="lineno">  793</span>&#160;          <span class="comment">// second point list</span></div>
<div class="line"><a name="l00794"></a><span class="lineno">  794</span>&#160;          <span class="keywordtype">int</span>,           <span class="comment">// stride</span></div>
<div class="line"><a name="l00795"></a><span class="lineno">  795</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span>*  <span class="comment">// point</span></div>
<div class="line"><a name="l00796"></a><span class="lineno">  796</span>&#160;          );</div>
<div class="line"><a name="l00797"></a><span class="lineno">  797</span>&#160; </div>
<div class="line"><a name="l00798"></a><span class="lineno">  798</span>&#160;ON_DECL</div>
<div class="line"><a name="l00799"></a><span class="lineno">  799</span>&#160;ON_BOOL32 ON_IsPointListClosed(</div>
<div class="line"><a name="l00800"></a><span class="lineno">  800</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00801"></a><span class="lineno">  801</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// true for homogeneos rational points</span></div>
<div class="line"><a name="l00802"></a><span class="lineno">  802</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// count</span></div>
<div class="line"><a name="l00803"></a><span class="lineno">  803</span>&#160;       <span class="keywordtype">int</span>,  <span class="comment">// stride</span></div>
<div class="line"><a name="l00804"></a><span class="lineno">  804</span>&#160;       <span class="keyword">const</span> <span class="keywordtype">double</span>*</div>
<div class="line"><a name="l00805"></a><span class="lineno">  805</span>&#160;       );</div>
<div class="line"><a name="l00806"></a><span class="lineno">  806</span>&#160; </div>
<div class="line"><a name="l00807"></a><span class="lineno">  807</span>&#160;ON_DECL</div>
<div class="line"><a name="l00808"></a><span class="lineno">  808</span>&#160;ON_BOOL32 ON_IsPointGridClosed(</div>
<div class="line"><a name="l00809"></a><span class="lineno">  809</span>&#160;        <span class="keywordtype">int</span>,  <span class="comment">// dim</span></div>
<div class="line"><a name="l00810"></a><span class="lineno">  810</span>&#160;        ON_BOOL32, <span class="comment">// true for homogeneous rational points</span></div>
<div class="line"><a name="l00811"></a><span class="lineno">  811</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_count0, point_count1,</span></div>
<div class="line"><a name="l00812"></a><span class="lineno">  812</span>&#160;        <span class="keywordtype">int</span>, <span class="keywordtype">int</span>, <span class="comment">// point_stride0, point_stride1,</span></div>
<div class="line"><a name="l00813"></a><span class="lineno">  813</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*,</div>
<div class="line"><a name="l00814"></a><span class="lineno">  814</span>&#160;        <span class="keywordtype">int</span>       <span class="comment">// dir = 0 or 1</span></div>
<div class="line"><a name="l00815"></a><span class="lineno">  815</span>&#160;       );</div>
<div class="line"><a name="l00816"></a><span class="lineno">  816</span>&#160; </div>
<div class="line"><a name="l00817"></a><span class="lineno">  817</span>&#160;ON_DECL</div>
<div class="line"><a name="l00818"></a><span class="lineno">  818</span>&#160;<span class="keywordtype">int</span> ON_SolveQuadraticEquation( <span class="comment">// solve a*X^2 + b*X + c = 0</span></div>
<div class="line"><a name="l00819"></a><span class="lineno">  819</span>&#160;        <span class="comment">// returns 0: two distinct real roots (r0 &lt; r1)</span></div>
<div class="line"><a name="l00820"></a><span class="lineno">  820</span>&#160;        <span class="comment">//         1: one real root (r0 = r1)</span></div>
<div class="line"><a name="l00821"></a><span class="lineno">  821</span>&#160;        <span class="comment">//         2: two complex conjugate roots (r0 +/- (r1)*sqrt(-1))</span></div>
<div class="line"><a name="l00822"></a><span class="lineno">  822</span>&#160;        <span class="comment">//        -1: failure - a = 0, b != 0        (r0 = r1 = -c/b)</span></div>
<div class="line"><a name="l00823"></a><span class="lineno">  823</span>&#160;        <span class="comment">//        -2: failure - a = 0, b  = 0 c != 0 (r0 = r1 = 0.0)</span></div>
<div class="line"><a name="l00824"></a><span class="lineno">  824</span>&#160;        <span class="comment">//        -3: failure - a = 0, b  = 0 c  = 0 (r0 = r1 = 0.0)</span></div>
<div class="line"><a name="l00825"></a><span class="lineno">  825</span>&#160;       <span class="keywordtype">double</span>, <span class="keywordtype">double</span>, <span class="keywordtype">double</span>, <span class="comment">// a, b, c</span></div>
<div class="line"><a name="l00826"></a><span class="lineno">  826</span>&#160;       <span class="keywordtype">double</span>*, <span class="keywordtype">double</span>*        <span class="comment">// roots r0 and r1 returned here</span></div>
<div class="line"><a name="l00827"></a><span class="lineno">  827</span>&#160;       );</div>
<div class="line"><a name="l00828"></a><span class="lineno">  828</span>&#160; </div>
<div class="line"><a name="l00829"></a><span class="lineno">  829</span>&#160;ON_DECL</div>
<div class="line"><a name="l00830"></a><span class="lineno">  830</span>&#160;ON_BOOL32 ON_SolveTriDiagonal( <span class="comment">// solve TriDiagMatrix( a,b,c )*X = d</span></div>
<div class="line"><a name="l00831"></a><span class="lineno">  831</span>&#160;        <span class="keywordtype">int</span>,               <span class="comment">// dimension of d and X (&gt;=1)</span></div>
<div class="line"><a name="l00832"></a><span class="lineno">  832</span>&#160;        <span class="keywordtype">int</span>,               <span class="comment">// number of equations (&gt;=2)</span></div>
<div class="line"><a name="l00833"></a><span class="lineno">  833</span>&#160;        <span class="keywordtype">double</span>*,           <span class="comment">// a[n-1] = sub-diagonal (a is modified)</span></div>
<div class="line"><a name="l00834"></a><span class="lineno">  834</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*,     <span class="comment">// b[n] = diagonal</span></div>
<div class="line"><a name="l00835"></a><span class="lineno">  835</span>&#160;        <span class="keywordtype">double</span>*,           <span class="comment">// c[n-1] = supra-diagonal</span></div>
<div class="line"><a name="l00836"></a><span class="lineno">  836</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*,     <span class="comment">// d[n*dim]</span></div>
<div class="line"><a name="l00837"></a><span class="lineno">  837</span>&#160;        <span class="keywordtype">double</span>*            <span class="comment">// X[n*dim] = unknowns</span></div>
<div class="line"><a name="l00838"></a><span class="lineno">  838</span>&#160;        );</div>
<div class="line"><a name="l00839"></a><span class="lineno">  839</span>&#160; </div>
<div class="line"><a name="l00840"></a><span class="lineno">  840</span>&#160;<span class="comment">// returns rank - if rank != 2, system is under determined</span></div>
<div class="line"><a name="l00841"></a><span class="lineno">  841</span>&#160;<span class="comment">// If rank = 2, then solution to </span></div>
<div class="line"><a name="l00842"></a><span class="lineno">  842</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00843"></a><span class="lineno">  843</span>&#160;<span class="comment">//          a00*x0 + a01*x1 = b0, </span></div>
<div class="line"><a name="l00844"></a><span class="lineno">  844</span>&#160;<span class="comment">//          a10*x0 + a11*x1 = b1 </span></div>
<div class="line"><a name="l00845"></a><span class="lineno">  845</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00846"></a><span class="lineno">  846</span>&#160;<span class="comment">// is returned</span></div>
<div class="line"><a name="l00847"></a><span class="lineno">  847</span>&#160;ON_DECL</div>
<div class="line"><a name="l00848"></a><span class="lineno">  848</span>&#160;<span class="keywordtype">int</span> ON_Solve2x2( </div>
<div class="line"><a name="l00849"></a><span class="lineno">  849</span>&#160;        <span class="keywordtype">double</span>, <span class="keywordtype">double</span>,   <span class="comment">// a00 a01 = first row of 2x2 matrix</span></div>
<div class="line"><a name="l00850"></a><span class="lineno">  850</span>&#160;        <span class="keywordtype">double</span>, <span class="keywordtype">double</span>,   <span class="comment">// a10 a11 = second row of 2x2 matrix</span></div>
<div class="line"><a name="l00851"></a><span class="lineno">  851</span>&#160;        <span class="keywordtype">double</span>, <span class="keywordtype">double</span>,   <span class="comment">// b0 b1</span></div>
<div class="line"><a name="l00852"></a><span class="lineno">  852</span>&#160;        <span class="keywordtype">double</span>*, <span class="keywordtype">double</span>*, <span class="comment">// x0, x1 if not NULL, then solution is returned here</span></div>
<div class="line"><a name="l00853"></a><span class="lineno">  853</span>&#160;        <span class="keywordtype">double</span>*           <span class="comment">// if not NULL, then pivot_ratio returned here</span></div>
<div class="line"><a name="l00854"></a><span class="lineno">  854</span>&#160;        );</div>
<div class="line"><a name="l00855"></a><span class="lineno">  855</span>&#160; </div>
<div class="line"><a name="l00856"></a><span class="lineno">  856</span>&#160;<span class="comment">// Description:</span></div>
<div class="line"><a name="l00857"></a><span class="lineno">  857</span>&#160;<span class="comment">//   Solves a system of 3 linear equations and 2 unknowns.</span></div>
<div class="line"><a name="l00858"></a><span class="lineno">  858</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00859"></a><span class="lineno">  859</span>&#160;<span class="comment">//          x*col0[0] + y*col1[0] = d0</span></div>
<div class="line"><a name="l00860"></a><span class="lineno">  860</span>&#160;<span class="comment">//          x*col0[1] + y*col1[1] = d0</span></div>
<div class="line"><a name="l00861"></a><span class="lineno">  861</span>&#160;<span class="comment">//          x*col0[2] + y*col1[2] = d0</span></div>
<div class="line"><a name="l00862"></a><span class="lineno">  862</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00863"></a><span class="lineno">  863</span>&#160;<span class="comment">// Parameters:</span></div>
<div class="line"><a name="l00864"></a><span class="lineno">  864</span>&#160;<span class="comment">//   col0 - [in] coefficents for &quot;x&quot; unknown</span></div>
<div class="line"><a name="l00865"></a><span class="lineno">  865</span>&#160;<span class="comment">//   col1 - [in] coefficents for &quot;y&quot; unknown</span></div>
<div class="line"><a name="l00866"></a><span class="lineno">  866</span>&#160;<span class="comment">//   d0 - [in] constants</span></div>
<div class="line"><a name="l00867"></a><span class="lineno">  867</span>&#160;<span class="comment">//   d1 - [in]</span></div>
<div class="line"><a name="l00868"></a><span class="lineno">  868</span>&#160;<span class="comment">//   d2 - [in]</span></div>
<div class="line"><a name="l00869"></a><span class="lineno">  869</span>&#160;<span class="comment">//   x - [out]</span></div>
<div class="line"><a name="l00870"></a><span class="lineno">  870</span>&#160;<span class="comment">//   y - [out]</span></div>
<div class="line"><a name="l00871"></a><span class="lineno">  871</span>&#160;<span class="comment">//   error - [out]</span></div>
<div class="line"><a name="l00872"></a><span class="lineno">  872</span>&#160;<span class="comment">//   pivot_ratio - [out]</span></div>
<div class="line"><a name="l00873"></a><span class="lineno">  873</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00874"></a><span class="lineno">  874</span>&#160;<span class="comment">// Returns:</span></div>
<div class="line"><a name="l00875"></a><span class="lineno">  875</span>&#160;<span class="comment">//   rank of the system.  </span></div>
<div class="line"><a name="l00876"></a><span class="lineno">  876</span>&#160;<span class="comment">//   If rank != 2, system is under determined</span></div>
<div class="line"><a name="l00877"></a><span class="lineno">  877</span>&#160;<span class="comment">//   If rank = 2, then the solution is</span></div>
<div class="line"><a name="l00878"></a><span class="lineno">  878</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l00879"></a><span class="lineno">  879</span>&#160;<span class="comment">//         (*x)*[col0] + (*y)*[col1]</span></div>
<div class="line"><a name="l00880"></a><span class="lineno">  880</span>&#160;<span class="comment">//         + (*error)*((col0 X col1)/|col0 X col1|)</span></div>
<div class="line"><a name="l00881"></a><span class="lineno">  881</span>&#160;<span class="comment">//         = (d0,d1,d2).</span></div>
<div class="line"><a name="l00882"></a><span class="lineno">  882</span>&#160;ON_DECL</div>
<div class="line"><a name="l00883"></a><span class="lineno">  883</span>&#160;<span class="keywordtype">int</span> ON_Solve3x2( </div>
<div class="line"><a name="l00884"></a><span class="lineno">  884</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>[3], <span class="comment">// col0</span></div>
<div class="line"><a name="l00885"></a><span class="lineno">  885</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>[3], <span class="comment">// col1</span></div>
<div class="line"><a name="l00886"></a><span class="lineno">  886</span>&#160;        <span class="keywordtype">double</span>,  <span class="comment">// d0</span></div>
<div class="line"><a name="l00887"></a><span class="lineno">  887</span>&#160;        <span class="keywordtype">double</span>,  <span class="comment">// d1</span></div>
<div class="line"><a name="l00888"></a><span class="lineno">  888</span>&#160;        <span class="keywordtype">double</span>,  <span class="comment">// d2</span></div>
<div class="line"><a name="l00889"></a><span class="lineno">  889</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// x</span></div>
<div class="line"><a name="l00890"></a><span class="lineno">  890</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// y</span></div>
<div class="line"><a name="l00891"></a><span class="lineno">  891</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// error</span></div>
<div class="line"><a name="l00892"></a><span class="lineno">  892</span>&#160;        <span class="keywordtype">double</span>*  <span class="comment">// pivot_ratio</span></div>
<div class="line"><a name="l00893"></a><span class="lineno">  893</span>&#160;        );</div>
<div class="line"><a name="l00894"></a><span class="lineno">  894</span>&#160; </div>
<div class="line"><a name="l00895"></a><span class="lineno">  895</span>&#160;<span class="comment">/* </span></div>
<div class="line"><a name="l00896"></a><span class="lineno">  896</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00897"></a><span class="lineno">  897</span>&#160;<span class="comment">  Use Gauss-Jordan elimination with full pivoting to solve </span></div>
<div class="line"><a name="l00898"></a><span class="lineno">  898</span>&#160;<span class="comment">  a system of 3 linear equations and 3 unknowns(x,y,z)</span></div>
<div class="line"><a name="l00899"></a><span class="lineno">  899</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00900"></a><span class="lineno">  900</span>&#160;<span class="comment">        x*row0[0] + y*row0[1] + z*row0[2] = d0</span></div>
<div class="line"><a name="l00901"></a><span class="lineno">  901</span>&#160;<span class="comment">        x*row1[0] + y*row1[1] + z*row1[2] = d1</span></div>
<div class="line"><a name="l00902"></a><span class="lineno">  902</span>&#160;<span class="comment">        x*row2[0] + y*row2[1] + z*row2[2] = d2</span></div>
<div class="line"><a name="l00903"></a><span class="lineno">  903</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00904"></a><span class="lineno">  904</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00905"></a><span class="lineno">  905</span>&#160;<span class="comment">    row0 - [in] first row of 3x3 matrix</span></div>
<div class="line"><a name="l00906"></a><span class="lineno">  906</span>&#160;<span class="comment">    row1 - [in] second row of 3x3 matrix</span></div>
<div class="line"><a name="l00907"></a><span class="lineno">  907</span>&#160;<span class="comment">    row2 - [in] third row of 3x3 matrix</span></div>
<div class="line"><a name="l00908"></a><span class="lineno">  908</span>&#160;<span class="comment">    d0 - [in] </span></div>
<div class="line"><a name="l00909"></a><span class="lineno">  909</span>&#160;<span class="comment">    d1 - [in] </span></div>
<div class="line"><a name="l00910"></a><span class="lineno">  910</span>&#160;<span class="comment">    d2 - [in] (d0,d1,d2) right hand column of system</span></div>
<div class="line"><a name="l00911"></a><span class="lineno">  911</span>&#160;<span class="comment">    x_addr - [in] first unknown</span></div>
<div class="line"><a name="l00912"></a><span class="lineno">  912</span>&#160;<span class="comment">    y_addr - [in] second unknown</span></div>
<div class="line"><a name="l00913"></a><span class="lineno">  913</span>&#160;<span class="comment">    z_addr - [in] third unknown</span></div>
<div class="line"><a name="l00914"></a><span class="lineno">  914</span>&#160;<span class="comment">    pivot_ratio - [out] if not NULL, the pivot ration is </span></div>
<div class="line"><a name="l00915"></a><span class="lineno">  915</span>&#160;<span class="comment">         returned here.  If the pivot ratio is &quot;small&quot;,</span></div>
<div class="line"><a name="l00916"></a><span class="lineno">  916</span>&#160;<span class="comment">         then the matrix may be singular or ill </span></div>
<div class="line"><a name="l00917"></a><span class="lineno">  917</span>&#160;<span class="comment">         conditioned. You should test the results </span></div>
<div class="line"><a name="l00918"></a><span class="lineno">  918</span>&#160;<span class="comment">         before you use them.  &quot;Small&quot; depends on the</span></div>
<div class="line"><a name="l00919"></a><span class="lineno">  919</span>&#160;<span class="comment">         precision of the input coefficients and the</span></div>
<div class="line"><a name="l00920"></a><span class="lineno">  920</span>&#160;<span class="comment">         use of the solution.  If you can&#39;t figure out</span></div>
<div class="line"><a name="l00921"></a><span class="lineno">  921</span>&#160;<span class="comment">         what &quot;small&quot; means in your case, then you</span></div>
<div class="line"><a name="l00922"></a><span class="lineno">  922</span>&#160;<span class="comment">         must check the solution before you use it.</span></div>
<div class="line"><a name="l00923"></a><span class="lineno">  923</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00924"></a><span class="lineno">  924</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00925"></a><span class="lineno">  925</span>&#160;<span class="comment">    The rank of the 3x3 matrix (0,1,2, or 3)</span></div>
<div class="line"><a name="l00926"></a><span class="lineno">  926</span>&#160;<span class="comment">    If ON_Solve3x3() is successful (returns 3), then</span></div>
<div class="line"><a name="l00927"></a><span class="lineno">  927</span>&#160;<span class="comment">    the solution is returned in </span></div>
<div class="line"><a name="l00928"></a><span class="lineno">  928</span>&#160;<span class="comment">    (*x_addr, *y_addr, *z_addr)</span></div>
<div class="line"><a name="l00929"></a><span class="lineno">  929</span>&#160;<span class="comment">    and *pivot_ratio = min(|pivots|)/max(|pivots|).</span></div>
<div class="line"><a name="l00930"></a><span class="lineno">  930</span>&#160;<span class="comment">    If the return code is &lt; 3, then (0,0,0) is returned</span></div>
<div class="line"><a name="l00931"></a><span class="lineno">  931</span>&#160;<span class="comment">    as the &quot;solution&quot;.</span></div>
<div class="line"><a name="l00932"></a><span class="lineno">  932</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00933"></a><span class="lineno">  933</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l00934"></a><span class="lineno">  934</span>&#160;<span class="comment">  ON_Solve2x2</span></div>
<div class="line"><a name="l00935"></a><span class="lineno">  935</span>&#160;<span class="comment">  ON_Solve3x2</span></div>
<div class="line"><a name="l00936"></a><span class="lineno">  936</span>&#160;<span class="comment">  ON_Solve4x4</span></div>
<div class="line"><a name="l00937"></a><span class="lineno">  937</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l00938"></a><span class="lineno">  938</span>&#160;ON_DECL</div>
<div class="line"><a name="l00939"></a><span class="lineno">  939</span>&#160;<span class="keywordtype">int</span> ON_Solve3x3( </div>
<div class="line"><a name="l00940"></a><span class="lineno">  940</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span> row0[3], </div>
<div class="line"><a name="l00941"></a><span class="lineno">  941</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span> row1[3], </div>
<div class="line"><a name="l00942"></a><span class="lineno">  942</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span> row2[3],</div>
<div class="line"><a name="l00943"></a><span class="lineno">  943</span>&#160;        <span class="keywordtype">double</span> d0, </div>
<div class="line"><a name="l00944"></a><span class="lineno">  944</span>&#160;        <span class="keywordtype">double</span> d1, </div>
<div class="line"><a name="l00945"></a><span class="lineno">  945</span>&#160;        <span class="keywordtype">double</span> d2,</div>
<div class="line"><a name="l00946"></a><span class="lineno">  946</span>&#160;        <span class="keywordtype">double</span>* x_addr, </div>
<div class="line"><a name="l00947"></a><span class="lineno">  947</span>&#160;        <span class="keywordtype">double</span>* y_addr, </div>
<div class="line"><a name="l00948"></a><span class="lineno">  948</span>&#160;        <span class="keywordtype">double</span>* z_addr,</div>
<div class="line"><a name="l00949"></a><span class="lineno">  949</span>&#160;        <span class="keywordtype">double</span>* pivot_ratio</div>
<div class="line"><a name="l00950"></a><span class="lineno">  950</span>&#160;        );</div>
<div class="line"><a name="l00951"></a><span class="lineno">  951</span>&#160; </div>
<div class="line"><a name="l00952"></a><span class="lineno">  952</span>&#160;<span class="comment">/* </span></div>
<div class="line"><a name="l00953"></a><span class="lineno">  953</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l00954"></a><span class="lineno">  954</span>&#160;<span class="comment">  Use Gauss-Jordan elimination with full pivoting to solve </span></div>
<div class="line"><a name="l00955"></a><span class="lineno">  955</span>&#160;<span class="comment">  a system of 4 linear equations and 4 unknowns(x,y,z,w)</span></div>
<div class="line"><a name="l00956"></a><span class="lineno">  956</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00957"></a><span class="lineno">  957</span>&#160;<span class="comment">        x*row0[0] + y*row0[1] + z*row0[2] + w*row0[3] = d0</span></div>
<div class="line"><a name="l00958"></a><span class="lineno">  958</span>&#160;<span class="comment">        x*row1[0] + y*row1[1] + z*row1[2] + w*row1[3] = d1</span></div>
<div class="line"><a name="l00959"></a><span class="lineno">  959</span>&#160;<span class="comment">        x*row2[0] + y*row2[1] + z*row2[2] + w*row2[3] = d2</span></div>
<div class="line"><a name="l00960"></a><span class="lineno">  960</span>&#160;<span class="comment">        x*row3[0] + y*row3[1] + z*row3[2] + w*row3[2] = d3</span></div>
<div class="line"><a name="l00961"></a><span class="lineno">  961</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00962"></a><span class="lineno">  962</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l00963"></a><span class="lineno">  963</span>&#160;<span class="comment">    row0 - [in] first row of 4x4 matrix</span></div>
<div class="line"><a name="l00964"></a><span class="lineno">  964</span>&#160;<span class="comment">    row1 - [in] second row of 4x4 matrix</span></div>
<div class="line"><a name="l00965"></a><span class="lineno">  965</span>&#160;<span class="comment">    row2 - [in] third row of 4x4 matrix</span></div>
<div class="line"><a name="l00966"></a><span class="lineno">  966</span>&#160;<span class="comment">    row3 - [in] forth row of 4x4 matrix</span></div>
<div class="line"><a name="l00967"></a><span class="lineno">  967</span>&#160;<span class="comment">    d0 - [in] </span></div>
<div class="line"><a name="l00968"></a><span class="lineno">  968</span>&#160;<span class="comment">    d1 - [in] </span></div>
<div class="line"><a name="l00969"></a><span class="lineno">  969</span>&#160;<span class="comment">    d2 - [in] </span></div>
<div class="line"><a name="l00970"></a><span class="lineno">  970</span>&#160;<span class="comment">    d3 - [in] (d0,d1,d2,d3) right hand column of system</span></div>
<div class="line"><a name="l00971"></a><span class="lineno">  971</span>&#160;<span class="comment">    x_addr - [in] first unknown</span></div>
<div class="line"><a name="l00972"></a><span class="lineno">  972</span>&#160;<span class="comment">    y_addr - [in] second unknown</span></div>
<div class="line"><a name="l00973"></a><span class="lineno">  973</span>&#160;<span class="comment">    z_addr - [in] third unknown</span></div>
<div class="line"><a name="l00974"></a><span class="lineno">  974</span>&#160;<span class="comment">    w_addr - [in] forth unknown</span></div>
<div class="line"><a name="l00975"></a><span class="lineno">  975</span>&#160;<span class="comment">    pivot_ratio - [out] if not NULL, the pivot ration is </span></div>
<div class="line"><a name="l00976"></a><span class="lineno">  976</span>&#160;<span class="comment">         returned here.  If the pivot ratio is &quot;small&quot;,</span></div>
<div class="line"><a name="l00977"></a><span class="lineno">  977</span>&#160;<span class="comment">         then the matrix may be singular or ill </span></div>
<div class="line"><a name="l00978"></a><span class="lineno">  978</span>&#160;<span class="comment">         conditioned. You should test the results </span></div>
<div class="line"><a name="l00979"></a><span class="lineno">  979</span>&#160;<span class="comment">         before you use them.  &quot;Small&quot; depends on the</span></div>
<div class="line"><a name="l00980"></a><span class="lineno">  980</span>&#160;<span class="comment">         precision of the input coefficients and the</span></div>
<div class="line"><a name="l00981"></a><span class="lineno">  981</span>&#160;<span class="comment">         use of the solution.  If you can&#39;t figure out</span></div>
<div class="line"><a name="l00982"></a><span class="lineno">  982</span>&#160;<span class="comment">         what &quot;small&quot; means in your case, then you</span></div>
<div class="line"><a name="l00983"></a><span class="lineno">  983</span>&#160;<span class="comment">         must check the solution before you use it.</span></div>
<div class="line"><a name="l00984"></a><span class="lineno">  984</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00985"></a><span class="lineno">  985</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l00986"></a><span class="lineno">  986</span>&#160;<span class="comment">    The rank of the 4x4 matrix (0,1,2,3, or 4)</span></div>
<div class="line"><a name="l00987"></a><span class="lineno">  987</span>&#160;<span class="comment">    If ON_Solve4x4() is successful (returns 4), then</span></div>
<div class="line"><a name="l00988"></a><span class="lineno">  988</span>&#160;<span class="comment">    the solution is returned in </span></div>
<div class="line"><a name="l00989"></a><span class="lineno">  989</span>&#160;<span class="comment">    (*x_addr, *y_addr, *z_addr, *w_addr)</span></div>
<div class="line"><a name="l00990"></a><span class="lineno">  990</span>&#160;<span class="comment">    and *pivot_ratio = min(|pivots|)/max(|pivots|).</span></div>
<div class="line"><a name="l00991"></a><span class="lineno">  991</span>&#160;<span class="comment">    If the return code is &lt; 4, then, it a solution exists,</span></div>
<div class="line"><a name="l00992"></a><span class="lineno">  992</span>&#160;<span class="comment">    on is returned.  However YOU MUST CHECK THE SOLUTION</span></div>
<div class="line"><a name="l00993"></a><span class="lineno">  993</span>&#160;<span class="comment">    IF THE RETURN CODE IS &lt; 4.</span></div>
<div class="line"><a name="l00994"></a><span class="lineno">  994</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00995"></a><span class="lineno">  995</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l00996"></a><span class="lineno">  996</span>&#160;<span class="comment">  ON_Solve2x2</span></div>
<div class="line"><a name="l00997"></a><span class="lineno">  997</span>&#160;<span class="comment">  ON_Solve3x2</span></div>
<div class="line"><a name="l00998"></a><span class="lineno">  998</span>&#160;<span class="comment">  ON_Solve3x3</span></div>
<div class="line"><a name="l00999"></a><span class="lineno">  999</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01000"></a><span class="lineno"> 1000</span>&#160;ON_DECL</div>
<div class="line"><a name="l01001"></a><span class="lineno"> 1001</span>&#160;<span class="keywordtype">int</span></div>
<div class="line"><a name="l01002"></a><span class="lineno"> 1002</span>&#160;ON_Solve4x4(</div>
<div class="line"><a name="l01003"></a><span class="lineno"> 1003</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span> row0[4], </div>
<div class="line"><a name="l01004"></a><span class="lineno"> 1004</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span> row1[4], </div>
<div class="line"><a name="l01005"></a><span class="lineno"> 1005</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span> row2[4],  </div>
<div class="line"><a name="l01006"></a><span class="lineno"> 1006</span>&#160;          <span class="keyword">const</span> <span class="keywordtype">double</span> row3[4],</div>
<div class="line"><a name="l01007"></a><span class="lineno"> 1007</span>&#160;          <span class="keywordtype">double</span> d0, </div>
<div class="line"><a name="l01008"></a><span class="lineno"> 1008</span>&#160;          <span class="keywordtype">double</span> d1, </div>
<div class="line"><a name="l01009"></a><span class="lineno"> 1009</span>&#160;          <span class="keywordtype">double</span> d2, </div>
<div class="line"><a name="l01010"></a><span class="lineno"> 1010</span>&#160;          <span class="keywordtype">double</span> d3,</div>
<div class="line"><a name="l01011"></a><span class="lineno"> 1011</span>&#160;          <span class="keywordtype">double</span>* x_addr, </div>
<div class="line"><a name="l01012"></a><span class="lineno"> 1012</span>&#160;          <span class="keywordtype">double</span>* y_addr, </div>
<div class="line"><a name="l01013"></a><span class="lineno"> 1013</span>&#160;          <span class="keywordtype">double</span>* z_addr, </div>
<div class="line"><a name="l01014"></a><span class="lineno"> 1014</span>&#160;          <span class="keywordtype">double</span>* w_addr,</div>
<div class="line"><a name="l01015"></a><span class="lineno"> 1015</span>&#160;          <span class="keywordtype">double</span>* pivot_ratio</div>
<div class="line"><a name="l01016"></a><span class="lineno"> 1016</span>&#160;          );</div>
<div class="line"><a name="l01017"></a><span class="lineno"> 1017</span>&#160; </div>
<div class="line"><a name="l01018"></a><span class="lineno"> 1018</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01019"></a><span class="lineno"> 1019</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01020"></a><span class="lineno"> 1020</span>&#160;<span class="comment">  Use Gauss-Jordan elimination to find a numerical </span></div>
<div class="line"><a name="l01021"></a><span class="lineno"> 1021</span>&#160;<span class="comment">  solution to M*X = B where M is a n x n matrix,</span></div>
<div class="line"><a name="l01022"></a><span class="lineno"> 1022</span>&#160;<span class="comment">  B is a known n-dimensional vector and X is</span></div>
<div class="line"><a name="l01023"></a><span class="lineno"> 1023</span>&#160;<span class="comment">  an unknown.</span></div>
<div class="line"><a name="l01024"></a><span class="lineno"> 1024</span>&#160;<span class="comment">Paramters:</span></div>
<div class="line"><a name="l01025"></a><span class="lineno"> 1025</span>&#160;<span class="comment">  bFullPivot - [in] if true, full pivoting is used,</span></div>
<div class="line"><a name="l01026"></a><span class="lineno"> 1026</span>&#160;<span class="comment">    otherwise partial pivoting is used.  In rare</span></div>
<div class="line"><a name="l01027"></a><span class="lineno"> 1027</span>&#160;<span class="comment">    cases full pivoting can produce a more accurate</span></div>
<div class="line"><a name="l01028"></a><span class="lineno"> 1028</span>&#160;<span class="comment">    answer and never produces a less accurate answer.</span></div>
<div class="line"><a name="l01029"></a><span class="lineno"> 1029</span>&#160;<span class="comment">    However full pivoting is slower.  If speed is an</span></div>
<div class="line"><a name="l01030"></a><span class="lineno"> 1030</span>&#160;<span class="comment">    issue, then experiement with bFullPivot=false</span></div>
<div class="line"><a name="l01031"></a><span class="lineno"> 1031</span>&#160;<span class="comment">    and see if it makes a difference.  Otherwise,</span></div>
<div class="line"><a name="l01032"></a><span class="lineno"> 1032</span>&#160;<span class="comment">    set it to true.</span></div>
<div class="line"><a name="l01033"></a><span class="lineno"> 1033</span>&#160;<span class="comment">  bNormalize - [in]</span></div>
<div class="line"><a name="l01034"></a><span class="lineno"> 1034</span>&#160;<span class="comment">    If bNormalize is true, then the rows of the</span></div>
<div class="line"><a name="l01035"></a><span class="lineno"> 1035</span>&#160;<span class="comment">    matrix are scaled so the sum of their squares</span></div>
<div class="line"><a name="l01036"></a><span class="lineno"> 1036</span>&#160;<span class="comment">    is one.  This doesn&#39;t make the solution more</span></div>
<div class="line"><a name="l01037"></a><span class="lineno"> 1037</span>&#160;<span class="comment">    accurate but in some cases it makes the pivot</span></div>
<div class="line"><a name="l01038"></a><span class="lineno"> 1038</span>&#160;<span class="comment">    ratio more meaningful.  Set bNormalize to</span></div>
<div class="line"><a name="l01039"></a><span class="lineno"> 1039</span>&#160;<span class="comment">    false unless you have a reason for setting it</span></div>
<div class="line"><a name="l01040"></a><span class="lineno"> 1040</span>&#160;<span class="comment">    to true.</span></div>
<div class="line"><a name="l01041"></a><span class="lineno"> 1041</span>&#160;<span class="comment">  n - [in] size of the matrix and vectors.</span></div>
<div class="line"><a name="l01042"></a><span class="lineno"> 1042</span>&#160;<span class="comment">  M - [in] n x n matrix.  The values in M are</span></div>
<div class="line"><a name="l01043"></a><span class="lineno"> 1043</span>&#160;<span class="comment">    changed as the solution is calculated.</span></div>
<div class="line"><a name="l01044"></a><span class="lineno"> 1044</span>&#160;<span class="comment">    If you need to preserve M for future use,</span></div>
<div class="line"><a name="l01045"></a><span class="lineno"> 1045</span>&#160;<span class="comment">    pass in a copy.</span></div>
<div class="line"><a name="l01046"></a><span class="lineno"> 1046</span>&#160;<span class="comment">  B - [in] n-dimensional vector.  The values in</span></div>
<div class="line"><a name="l01047"></a><span class="lineno"> 1047</span>&#160;<span class="comment">    B are changed as the solution is calculated.</span></div>
<div class="line"><a name="l01048"></a><span class="lineno"> 1048</span>&#160;<span class="comment">    If you need to preserve B for future use,</span></div>
<div class="line"><a name="l01049"></a><span class="lineno"> 1049</span>&#160;<span class="comment">    pass in a copy.</span></div>
<div class="line"><a name="l01050"></a><span class="lineno"> 1050</span>&#160;<span class="comment">  X - [out] solution to M*X = B.</span></div>
<div class="line"><a name="l01051"></a><span class="lineno"> 1051</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01052"></a><span class="lineno"> 1052</span>&#160;<span class="comment">  If the returned value is &lt;= 0.0, the input matrix</span></div>
<div class="line"><a name="l01053"></a><span class="lineno"> 1053</span>&#160;<span class="comment">  has rank &lt; n and no solution is returned in X.</span></div>
<div class="line"><a name="l01054"></a><span class="lineno"> 1054</span>&#160;<span class="comment">  If the returned value is &gt; 0.0, then a solution is</span></div>
<div class="line"><a name="l01055"></a><span class="lineno"> 1055</span>&#160;<span class="comment">  returned in X and the returned value is the ratio</span></div>
<div class="line"><a name="l01056"></a><span class="lineno"> 1056</span>&#160;<span class="comment">  (minimum pivot)/(maximum pivot).  This value is</span></div>
<div class="line"><a name="l01057"></a><span class="lineno"> 1057</span>&#160;<span class="comment">  called the pivot ratio and will be denoted &quot;pr&quot;</span></div>
<div class="line"><a name="l01058"></a><span class="lineno"> 1058</span>&#160;<span class="comment">  the discussion below. If pr &lt;= 1e-15, then</span></div>
<div class="line"><a name="l01059"></a><span class="lineno"> 1059</span>&#160;<span class="comment">  M was nearly degenerate and the solution should be</span></div>
<div class="line"><a name="l01060"></a><span class="lineno"> 1060</span>&#160;<span class="comment">  used with caution.  If an accurate solution is</span></div>
<div class="line"><a name="l01061"></a><span class="lineno"> 1061</span>&#160;<span class="comment">  critcial, then check the solution anytime pr &lt;= 1e-10</span></div>
<div class="line"><a name="l01062"></a><span class="lineno"> 1062</span>&#160;<span class="comment">  In general, the difference between M*X and B will be</span></div>
<div class="line"><a name="l01063"></a><span class="lineno"> 1063</span>&#160;<span class="comment">  reasonably small.  However, when the pr is small</span></div>
<div class="line"><a name="l01064"></a><span class="lineno"> 1064</span>&#160;<span class="comment">  there tend to be vector E, substantually different</span></div>
<div class="line"><a name="l01065"></a><span class="lineno"> 1065</span>&#160;<span class="comment">  from zero, such that M*(X+E) - B is also reasonably</span></div>
<div class="line"><a name="l01066"></a><span class="lineno"> 1066</span>&#160;<span class="comment">  small.</span></div>
<div class="line"><a name="l01067"></a><span class="lineno"> 1067</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l01068"></a><span class="lineno"> 1068</span>&#160;<span class="comment">  ON_Solve2x2</span></div>
<div class="line"><a name="l01069"></a><span class="lineno"> 1069</span>&#160;<span class="comment">  ON_Solve3x3</span></div>
<div class="line"><a name="l01070"></a><span class="lineno"> 1070</span>&#160;<span class="comment">  ON_Solve4x4</span></div>
<div class="line"><a name="l01071"></a><span class="lineno"> 1071</span>&#160;<span class="comment">  ON_Solve3x2</span></div>
<div class="line"><a name="l01072"></a><span class="lineno"> 1072</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01073"></a><span class="lineno"> 1073</span>&#160;ON_DECL</div>
<div class="line"><a name="l01074"></a><span class="lineno"> 1074</span>&#160;<span class="keywordtype">double</span> ON_SolveNxN(<span class="keywordtype">bool</span> bFullPivot, <span class="keywordtype">bool</span> bNormalize, <span class="keywordtype">int</span> n, <span class="keywordtype">double</span>* M[], <span class="keywordtype">double</span> B[], <span class="keywordtype">double</span> X[]);</div>
<div class="line"><a name="l01075"></a><span class="lineno"> 1075</span>&#160; </div>
<div class="line"><a name="l01076"></a><span class="lineno"> 1076</span>&#160; </div>
<div class="line"><a name="l01077"></a><span class="lineno"> 1077</span>&#160;<span class="comment">// return false if determinant is (nearly) singular</span></div>
<div class="line"><a name="l01078"></a><span class="lineno"> 1078</span>&#160;ON_DECL</div>
<div class="line"><a name="l01079"></a><span class="lineno"> 1079</span>&#160;ON_BOOL32 ON_EvJacobian( </div>
<div class="line"><a name="l01080"></a><span class="lineno"> 1080</span>&#160;        <span class="keywordtype">double</span>, <span class="comment">// ds o ds</span></div>
<div class="line"><a name="l01081"></a><span class="lineno"> 1081</span>&#160;        <span class="keywordtype">double</span>, <span class="comment">// ds o dt</span></div>
<div class="line"><a name="l01082"></a><span class="lineno"> 1082</span>&#160;        <span class="keywordtype">double</span>, <span class="comment">// dt o dt</span></div>
<div class="line"><a name="l01083"></a><span class="lineno"> 1083</span>&#160;        <span class="keywordtype">double</span>* <span class="comment">// jacobian = determinant ( ds_o_ds dt_o_dt / ds_o_dt ds_o_dt )</span></div>
<div class="line"><a name="l01084"></a><span class="lineno"> 1084</span>&#160;        );</div>
<div class="line"><a name="l01085"></a><span class="lineno"> 1085</span>&#160; </div>
<div class="line"><a name="l01086"></a><span class="lineno"> 1086</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01087"></a><span class="lineno"> 1087</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01088"></a><span class="lineno"> 1088</span>&#160;<span class="comment">  Finds scalars x and y so that the component of V in the plane</span></div>
<div class="line"><a name="l01089"></a><span class="lineno"> 1089</span>&#160;<span class="comment">  of A and B is x*A + y*B.</span></div>
<div class="line"><a name="l01090"></a><span class="lineno"> 1090</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01091"></a><span class="lineno"> 1091</span>&#160;<span class="comment">  V - [in]</span></div>
<div class="line"><a name="l01092"></a><span class="lineno"> 1092</span>&#160;<span class="comment">  A - [in] nonzero and not parallel to B</span></div>
<div class="line"><a name="l01093"></a><span class="lineno"> 1093</span>&#160;<span class="comment">  B - [in] nonzero and not parallel to A</span></div>
<div class="line"><a name="l01094"></a><span class="lineno"> 1094</span>&#160;<span class="comment">  x - [out]</span></div>
<div class="line"><a name="l01095"></a><span class="lineno"> 1095</span>&#160;<span class="comment">  y - [out]</span></div>
<div class="line"><a name="l01096"></a><span class="lineno"> 1096</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01097"></a><span class="lineno"> 1097</span>&#160;<span class="comment">  1 - The rank of the problem is 2.  The decomposition is unique.</span></div>
<div class="line"><a name="l01098"></a><span class="lineno"> 1098</span>&#160;<span class="comment">    0 - The rank less than 2.  Either there is no solution or there</span></div>
<div class="line"><a name="l01099"></a><span class="lineno"> 1099</span>&#160;<span class="comment">            are infinitely many solutions.</span></div>
<div class="line"><a name="l01100"></a><span class="lineno"> 1100</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l01101"></a><span class="lineno"> 1101</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l01102"></a><span class="lineno"> 1102</span>&#160;<span class="comment">  ON_Solve2x2</span></div>
<div class="line"><a name="l01103"></a><span class="lineno"> 1103</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01104"></a><span class="lineno"> 1104</span>&#160;ON_DECL</div>
<div class="line"><a name="l01105"></a><span class="lineno"> 1105</span>&#160;<span class="keywordtype">int</span> ON_DecomposeVector(</div>
<div class="line"><a name="l01106"></a><span class="lineno"> 1106</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; V,</div>
<div class="line"><a name="l01107"></a><span class="lineno"> 1107</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; A,</div>
<div class="line"><a name="l01108"></a><span class="lineno"> 1108</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; B,</div>
<div class="line"><a name="l01109"></a><span class="lineno"> 1109</span>&#160;        <span class="keywordtype">double</span>* x, <span class="keywordtype">double</span>* y</div>
<div class="line"><a name="l01110"></a><span class="lineno"> 1110</span>&#160;        );</div>
<div class="line"><a name="l01111"></a><span class="lineno"> 1111</span>&#160; </div>
<div class="line"><a name="l01112"></a><span class="lineno"> 1112</span>&#160; </div>
<div class="line"><a name="l01113"></a><span class="lineno"> 1113</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01114"></a><span class="lineno"> 1114</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01115"></a><span class="lineno"> 1115</span>&#160;<span class="comment">   Evaluate partial derivatives of surface unit normal</span></div>
<div class="line"><a name="l01116"></a><span class="lineno"> 1116</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01117"></a><span class="lineno"> 1117</span>&#160;<span class="comment">  ds - [in]</span></div>
<div class="line"><a name="l01118"></a><span class="lineno"> 1118</span>&#160;<span class="comment">  dt - [in] surface first partial derivatives</span></div>
<div class="line"><a name="l01119"></a><span class="lineno"> 1119</span>&#160;<span class="comment">  dss - [in]</span></div>
<div class="line"><a name="l01120"></a><span class="lineno"> 1120</span>&#160;<span class="comment">  dst - [in]</span></div>
<div class="line"><a name="l01121"></a><span class="lineno"> 1121</span>&#160;<span class="comment">  dtt - [in] surface second partial derivatives</span></div>
<div class="line"><a name="l01122"></a><span class="lineno"> 1122</span>&#160;<span class="comment">  ns - [out]</span></div>
<div class="line"><a name="l01123"></a><span class="lineno"> 1123</span>&#160;<span class="comment">  nt - [out] First partial derivatives of surface unit normal</span></div>
<div class="line"><a name="l01124"></a><span class="lineno"> 1124</span>&#160;<span class="comment">             (If the Jacobian is degenerate, ns and nt are set to zero.)</span></div>
<div class="line"><a name="l01125"></a><span class="lineno"> 1125</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01126"></a><span class="lineno"> 1126</span>&#160;<span class="comment">  true if Jacobian is nondegenerate</span></div>
<div class="line"><a name="l01127"></a><span class="lineno"> 1127</span>&#160;<span class="comment">  false if Jacobian is degenerate</span></div>
<div class="line"><a name="l01128"></a><span class="lineno"> 1128</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01129"></a><span class="lineno"> 1129</span>&#160;ON_DECL</div>
<div class="line"><a name="l01130"></a><span class="lineno"> 1130</span>&#160;ON_BOOL32 ON_EvNormalPartials(</div>
<div class="line"><a name="l01131"></a><span class="lineno"> 1131</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; ds,</div>
<div class="line"><a name="l01132"></a><span class="lineno"> 1132</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; dt,</div>
<div class="line"><a name="l01133"></a><span class="lineno"> 1133</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; dss,</div>
<div class="line"><a name="l01134"></a><span class="lineno"> 1134</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; dst,</div>
<div class="line"><a name="l01135"></a><span class="lineno"> 1135</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; dtt,</div>
<div class="line"><a name="l01136"></a><span class="lineno"> 1136</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; ns,</div>
<div class="line"><a name="l01137"></a><span class="lineno"> 1137</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; nt</div>
<div class="line"><a name="l01138"></a><span class="lineno"> 1138</span>&#160;        );</div>
<div class="line"><a name="l01139"></a><span class="lineno"> 1139</span>&#160; </div>
<div class="line"><a name="l01140"></a><span class="lineno"> 1140</span>&#160;ON_DECL</div>
<div class="line"><a name="l01141"></a><span class="lineno"> 1141</span>&#160;ON_BOOL32 </div>
<div class="line"><a name="l01142"></a><span class="lineno"> 1142</span>&#160;ON_Pullback3dVector( <span class="comment">// use to pull 3d vector back to surface parameter space</span></div>
<div class="line"><a name="l01143"></a><span class="lineno"> 1143</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,   <span class="comment">// 3d vector</span></div>
<div class="line"><a name="l01144"></a><span class="lineno"> 1144</span>&#160;      <span class="keywordtype">double</span>,              <span class="comment">// signed distance from vector location to closet point on surface</span></div>
<div class="line"><a name="l01145"></a><span class="lineno"> 1145</span>&#160;                                    <span class="comment">// &lt; 0 if point is below with respect to Du x Dv</span></div>
<div class="line"><a name="l01146"></a><span class="lineno"> 1146</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,     <span class="comment">// ds      surface first partials</span></div>
<div class="line"><a name="l01147"></a><span class="lineno"> 1147</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,     <span class="comment">// dt</span></div>
<div class="line"><a name="l01148"></a><span class="lineno"> 1148</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,     <span class="comment">// dss     surface 2nd partials</span></div>
<div class="line"><a name="l01149"></a><span class="lineno"> 1149</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,     <span class="comment">// dst     (used only when dist != 0)</span></div>
<div class="line"><a name="l01150"></a><span class="lineno"> 1150</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,     <span class="comment">// dtt</span></div>
<div class="line"><a name="l01151"></a><span class="lineno"> 1151</span>&#160;      <a class="code" href="class_o_n__2d_vector.html">ON_2dVector</a>&amp;            <span class="comment">// pullback</span></div>
<div class="line"><a name="l01152"></a><span class="lineno"> 1152</span>&#160;      );</div>
<div class="line"><a name="l01153"></a><span class="lineno"> 1153</span>&#160; </div>
<div class="line"><a name="l01154"></a><span class="lineno"> 1154</span>&#160;ON_DECL</div>
<div class="line"><a name="l01155"></a><span class="lineno"> 1155</span>&#160;ON_BOOL32 </div>
<div class="line"><a name="l01156"></a><span class="lineno"> 1156</span>&#160;ON_GetParameterTolerance(</div>
<div class="line"><a name="l01157"></a><span class="lineno"> 1157</span>&#160;        <span class="keywordtype">double</span>,   <span class="comment">// t0      domain</span></div>
<div class="line"><a name="l01158"></a><span class="lineno"> 1158</span>&#160;        <span class="keywordtype">double</span>,   <span class="comment">// t1 </span></div>
<div class="line"><a name="l01159"></a><span class="lineno"> 1159</span>&#160;        <span class="keywordtype">double</span>,   <span class="comment">// t       parameter in domain</span></div>
<div class="line"><a name="l01160"></a><span class="lineno"> 1160</span>&#160;        <span class="keywordtype">double</span>*,  <span class="comment">// tminus  parameter tolerance (tminus, tplus) returned here</span></div>
<div class="line"><a name="l01161"></a><span class="lineno"> 1161</span>&#160;        <span class="keywordtype">double</span>*   <span class="comment">// tplus</span></div>
<div class="line"><a name="l01162"></a><span class="lineno"> 1162</span>&#160;        );</div>
<div class="line"><a name="l01163"></a><span class="lineno"> 1163</span>&#160; </div>
<div class="line"><a name="l01164"></a><span class="lineno"> 1164</span>&#160; </div>
<div class="line"><a name="l01165"></a><span class="lineno"> 1165</span>&#160;ON_DECL</div>
<div class="line"><a name="l01166"></a><span class="lineno"> 1166</span>&#160;ON_BOOL32 ON_EvNormal(</div>
<div class="line"><a name="l01167"></a><span class="lineno"> 1167</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// limit_dir 0=default,1=from quadrant I, 2 = from quadrant II, ...</span></div>
<div class="line"><a name="l01168"></a><span class="lineno"> 1168</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// first partials (Du,Dv)</span></div>
<div class="line"><a name="l01169"></a><span class="lineno"> 1169</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// optional second partials (Duu, Duv, Dvv)</span></div>
<div class="line"><a name="l01170"></a><span class="lineno"> 1170</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; <span class="comment">// unit normal returned here</span></div>
<div class="line"><a name="l01171"></a><span class="lineno"> 1171</span>&#160;        );</div>
<div class="line"><a name="l01172"></a><span class="lineno"> 1172</span>&#160; </div>
<div class="line"><a name="l01173"></a><span class="lineno"> 1173</span>&#160;<span class="comment">// returns false if first returned tangent is zero</span></div>
<div class="line"><a name="l01174"></a><span class="lineno"> 1174</span>&#160;ON_DECL</div>
<div class="line"><a name="l01175"></a><span class="lineno"> 1175</span>&#160;<span class="keywordtype">bool</span> ON_EvTangent(</div>
<div class="line"><a name="l01176"></a><span class="lineno"> 1176</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// first derivative</span></div>
<div class="line"><a name="l01177"></a><span class="lineno"> 1177</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// second derivative</span></div>
<div class="line"><a name="l01178"></a><span class="lineno"> 1178</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;        <span class="comment">// Unit tangent returned here</span></div>
<div class="line"><a name="l01179"></a><span class="lineno"> 1179</span>&#160;        );</div>
<div class="line"><a name="l01180"></a><span class="lineno"> 1180</span>&#160; </div>
<div class="line"><a name="l01181"></a><span class="lineno"> 1181</span>&#160;<span class="comment">// returns false if first derivtive is zero</span></div>
<div class="line"><a name="l01182"></a><span class="lineno"> 1182</span>&#160;ON_DECL</div>
<div class="line"><a name="l01183"></a><span class="lineno"> 1183</span>&#160;ON_BOOL32 ON_EvCurvature(</div>
<div class="line"><a name="l01184"></a><span class="lineno"> 1184</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// first derivative</span></div>
<div class="line"><a name="l01185"></a><span class="lineno"> 1185</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// second derivative</span></div>
<div class="line"><a name="l01186"></a><span class="lineno"> 1186</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;,       <span class="comment">// Unit tangent returned here</span></div>
<div class="line"><a name="l01187"></a><span class="lineno"> 1187</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;        <span class="comment">// Curvature returned here</span></div>
<div class="line"><a name="l01188"></a><span class="lineno"> 1188</span>&#160;        );</div>
<div class="line"><a name="l01189"></a><span class="lineno"> 1189</span>&#160; </div>
<div class="line"><a name="l01190"></a><span class="lineno"> 1190</span>&#160;ON_DECL</div>
<div class="line"><a name="l01191"></a><span class="lineno"> 1191</span>&#160;ON_BOOL32 ON_EvPrincipalCurvatures( </div>
<div class="line"><a name="l01192"></a><span class="lineno"> 1192</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Ds,</span></div>
<div class="line"><a name="l01193"></a><span class="lineno"> 1193</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Dt,</span></div>
<div class="line"><a name="l01194"></a><span class="lineno"> 1194</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Dss,</span></div>
<div class="line"><a name="l01195"></a><span class="lineno"> 1195</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Dst,</span></div>
<div class="line"><a name="l01196"></a><span class="lineno"> 1196</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Dtt,</span></div>
<div class="line"><a name="l01197"></a><span class="lineno"> 1197</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// N,   // unit normal to surface (use ON_EvNormal())</span></div>
<div class="line"><a name="l01198"></a><span class="lineno"> 1198</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// gauss,  // = Gaussian curvature = kappa1*kappa2</span></div>
<div class="line"><a name="l01199"></a><span class="lineno"> 1199</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// mean,   // = mean curvature = (kappa1+kappa2)/2</span></div>
<div class="line"><a name="l01200"></a><span class="lineno"> 1200</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// kappa1, // = largest principal curvature value (may be negative)</span></div>
<div class="line"><a name="l01201"></a><span class="lineno"> 1201</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// kappa2, // = smallest principal curvature value (may be negative)</span></div>
<div class="line"><a name="l01202"></a><span class="lineno"> 1202</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// K1,     // kappa1 unit principal curvature direction</span></div>
<div class="line"><a name="l01203"></a><span class="lineno"> 1203</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;  <span class="comment">// K2      // kappa2 unit principal curvature direction</span></div>
<div class="line"><a name="l01204"></a><span class="lineno"> 1204</span>&#160;                        <span class="comment">// output K1,K2,N is right handed frame</span></div>
<div class="line"><a name="l01205"></a><span class="lineno"> 1205</span>&#160;        );</div>
<div class="line"><a name="l01206"></a><span class="lineno"> 1206</span>&#160; </div>
<div class="line"><a name="l01207"></a><span class="lineno"> 1207</span>&#160;ON_DECL</div>
<div class="line"><a name="l01208"></a><span class="lineno"> 1208</span>&#160;ON_BOOL32 ON_EvPrincipalCurvatures( </div>
<div class="line"><a name="l01209"></a><span class="lineno"> 1209</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Ds,</span></div>
<div class="line"><a name="l01210"></a><span class="lineno"> 1210</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// Dt,</span></div>
<div class="line"><a name="l01211"></a><span class="lineno"> 1211</span>&#160;        <span class="keywordtype">double</span> l, <span class="comment">// Dss*N Second fundamental form coefficients</span></div>
<div class="line"><a name="l01212"></a><span class="lineno"> 1212</span>&#160;        <span class="keywordtype">double</span> m, <span class="comment">// Dst*N,</span></div>
<div class="line"><a name="l01213"></a><span class="lineno"> 1213</span>&#160;        <span class="keywordtype">double</span> n, <span class="comment">// Dtt*N,</span></div>
<div class="line"><a name="l01214"></a><span class="lineno"> 1214</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// N,   // unit normal to surface (use ON_EvNormal())</span></div>
<div class="line"><a name="l01215"></a><span class="lineno"> 1215</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// gauss,  // = Gaussian curvature = kappa1*kappa2</span></div>
<div class="line"><a name="l01216"></a><span class="lineno"> 1216</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// mean,   // = mean curvature = (kappa1+kappa2)/2</span></div>
<div class="line"><a name="l01217"></a><span class="lineno"> 1217</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// kappa1, // = largest principal curvature value (may be negative)</span></div>
<div class="line"><a name="l01218"></a><span class="lineno"> 1218</span>&#160;        <span class="keywordtype">double</span>*, <span class="comment">// kappa2, // = smallest principal curvature value (may be negative)</span></div>
<div class="line"><a name="l01219"></a><span class="lineno"> 1219</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// K1,     // kappa1 unit principal curvature direction</span></div>
<div class="line"><a name="l01220"></a><span class="lineno"> 1220</span>&#160;        <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;  <span class="comment">// K2      // kappa2 unit principal curvature direction</span></div>
<div class="line"><a name="l01221"></a><span class="lineno"> 1221</span>&#160;                        <span class="comment">// output K1,K2,N is right handed frame</span></div>
<div class="line"><a name="l01222"></a><span class="lineno"> 1222</span>&#160;        );</div>
<div class="line"><a name="l01223"></a><span class="lineno"> 1223</span>&#160; </div>
<div class="line"><a name="l01224"></a><span class="lineno"> 1224</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01225"></a><span class="lineno"> 1225</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01226"></a><span class="lineno"> 1226</span>&#160;<span class="comment">  Evaluate sectional curvature from surface derivatives and </span></div>
<div class="line"><a name="l01227"></a><span class="lineno"> 1227</span>&#160;<span class="comment">  section plane normal.</span></div>
<div class="line"><a name="l01228"></a><span class="lineno"> 1228</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01229"></a><span class="lineno"> 1229</span>&#160;<span class="comment">  S10, S01 - [in]</span></div>
<div class="line"><a name="l01230"></a><span class="lineno"> 1230</span>&#160;<span class="comment">    surface 1st partial derivatives</span></div>
<div class="line"><a name="l01231"></a><span class="lineno"> 1231</span>&#160;<span class="comment">  S20, S11, S02 - [in]</span></div>
<div class="line"><a name="l01232"></a><span class="lineno"> 1232</span>&#160;<span class="comment">    surface 2nd partial derivatives</span></div>
<div class="line"><a name="l01233"></a><span class="lineno"> 1233</span>&#160;<span class="comment">  planeNormal - [in]</span></div>
<div class="line"><a name="l01234"></a><span class="lineno"> 1234</span>&#160;<span class="comment">    unit normal to section plane</span></div>
<div class="line"><a name="l01235"></a><span class="lineno"> 1235</span>&#160;<span class="comment">  K - [out] Sectional curvature</span></div>
<div class="line"><a name="l01236"></a><span class="lineno"> 1236</span>&#160;<span class="comment">    Curvature of the intersection curve of the surface</span></div>
<div class="line"><a name="l01237"></a><span class="lineno"> 1237</span>&#160;<span class="comment">    and plane through the surface point where the partial</span></div>
<div class="line"><a name="l01238"></a><span class="lineno"> 1238</span>&#160;<span class="comment">    derivatives were evaluationed.</span></div>
<div class="line"><a name="l01239"></a><span class="lineno"> 1239</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01240"></a><span class="lineno"> 1240</span>&#160;<span class="comment">  True if successful.</span></div>
<div class="line"><a name="l01241"></a><span class="lineno"> 1241</span>&#160;<span class="comment">  False if first partials are not linearly independent, in</span></div>
<div class="line"><a name="l01242"></a><span class="lineno"> 1242</span>&#160;<span class="comment">  which case the K is set to zero.</span></div>
<div class="line"><a name="l01243"></a><span class="lineno"> 1243</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01244"></a><span class="lineno"> 1244</span>&#160;ON_DECL</div>
<div class="line"><a name="l01245"></a><span class="lineno"> 1245</span>&#160;<span class="keywordtype">bool</span> ON_EvSectionalCurvature( </div>
<div class="line"><a name="l01246"></a><span class="lineno"> 1246</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; S10, </div>
<div class="line"><a name="l01247"></a><span class="lineno"> 1247</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; S01,</div>
<div class="line"><a name="l01248"></a><span class="lineno"> 1248</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; S20, </div>
<div class="line"><a name="l01249"></a><span class="lineno"> 1249</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; S11, </div>
<div class="line"><a name="l01250"></a><span class="lineno"> 1250</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; S02,</div>
<div class="line"><a name="l01251"></a><span class="lineno"> 1251</span>&#160;    <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; planeNormal,</div>
<div class="line"><a name="l01252"></a><span class="lineno"> 1252</span>&#160;    <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp; K </div>
<div class="line"><a name="l01253"></a><span class="lineno"> 1253</span>&#160;    );</div>
<div class="line"><a name="l01254"></a><span class="lineno"> 1254</span>&#160; </div>
<div class="line"><a name="l01255"></a><span class="lineno"> 1255</span>&#160; </div>
<div class="line"><a name="l01256"></a><span class="lineno"> 1256</span>&#160;ON_DECL</div>
<div class="line"><a name="l01257"></a><span class="lineno"> 1257</span>&#160;<a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> ON_NormalCurvature( </div>
<div class="line"><a name="l01258"></a><span class="lineno"> 1258</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface 1rst partial (Ds)</span></div>
<div class="line"><a name="l01259"></a><span class="lineno"> 1259</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface 1rst partial (Dt)</span></div>
<div class="line"><a name="l01260"></a><span class="lineno"> 1260</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface 1rst partial (Dss)</span></div>
<div class="line"><a name="l01261"></a><span class="lineno"> 1261</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface 1rst partial (Dst)</span></div>
<div class="line"><a name="l01262"></a><span class="lineno"> 1262</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface 1rst partial (Dtt)</span></div>
<div class="line"><a name="l01263"></a><span class="lineno"> 1263</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;, <span class="comment">// surface unit normal</span></div>
<div class="line"><a name="l01264"></a><span class="lineno"> 1264</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a>&amp;  <span class="comment">// unit tangent direction</span></div>
<div class="line"><a name="l01265"></a><span class="lineno"> 1265</span>&#160;        );</div>
<div class="line"><a name="l01266"></a><span class="lineno"> 1266</span>&#160; </div>
<div class="line"><a name="l01267"></a><span class="lineno"> 1267</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01268"></a><span class="lineno"> 1268</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01269"></a><span class="lineno"> 1269</span>&#160;<span class="comment">  Determing if two curvatrues are different enough</span></div>
<div class="line"><a name="l01270"></a><span class="lineno"> 1270</span>&#160;<span class="comment">  to qualify as a curvature discontinuity.</span></div>
<div class="line"><a name="l01271"></a><span class="lineno"> 1271</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01272"></a><span class="lineno"> 1272</span>&#160;<span class="comment">  Km - [in]</span></div>
<div class="line"><a name="l01273"></a><span class="lineno"> 1273</span>&#160;<span class="comment">  Kp - [in]</span></div>
<div class="line"><a name="l01274"></a><span class="lineno"> 1274</span>&#160;<span class="comment">    Km and Kp should be curvatures evaluated at the same</span></div>
<div class="line"><a name="l01275"></a><span class="lineno"> 1275</span>&#160;<span class="comment">    parameters using limits from below (minus) and above (plus).</span></div>
<div class="line"><a name="l01276"></a><span class="lineno"> 1276</span>&#160;<span class="comment">    The assumption is that you have already compared the</span></div>
<div class="line"><a name="l01277"></a><span class="lineno"> 1277</span>&#160;<span class="comment">    points and tangents and consider to curve to be G1 at the</span></div>
<div class="line"><a name="l01278"></a><span class="lineno"> 1278</span>&#160;<span class="comment">    point in question.</span></div>
<div class="line"><a name="l01279"></a><span class="lineno"> 1279</span>&#160;<span class="comment">  cos_angle_tolerance - [in]</span></div>
<div class="line"><a name="l01280"></a><span class="lineno"> 1280</span>&#160;<span class="comment">    If the inut value of cos_angle_tolerance &gt;= -1.0</span></div>
<div class="line"><a name="l01281"></a><span class="lineno"> 1281</span>&#160;<span class="comment">    and cos_angle_tolerance &lt;= 1.0 and</span></div>
<div class="line"><a name="l01282"></a><span class="lineno"> 1282</span>&#160;<span class="comment">    Km o Kp &lt; cos_angle_tolerance*|Km|*|Kp|, then</span></div>
<div class="line"><a name="l01283"></a><span class="lineno"> 1283</span>&#160;<span class="comment">    true is returned.  Otherwise it is assumed Km and Kp</span></div>
<div class="line"><a name="l01284"></a><span class="lineno"> 1284</span>&#160;<span class="comment">    are parallel. If the curve being tested is nonplanar,</span></div>
<div class="line"><a name="l01285"></a><span class="lineno"> 1285</span>&#160;<span class="comment">    then use something like cos(2*tangent angle tolerance)</span></div>
<div class="line"><a name="l01286"></a><span class="lineno"> 1286</span>&#160;<span class="comment">    for this parameter. If the curve being tested is planar,</span></div>
<div class="line"><a name="l01287"></a><span class="lineno"> 1287</span>&#160;<span class="comment">    then 0.0 will work fine.</span></div>
<div class="line"><a name="l01288"></a><span class="lineno"> 1288</span>&#160;<span class="comment">  curvature_tolerance - [in]</span></div>
<div class="line"><a name="l01289"></a><span class="lineno"> 1289</span>&#160;<span class="comment">    If |Kp-Km| &lt;= curvature_tolerance,</span></div>
<div class="line"><a name="l01290"></a><span class="lineno"> 1290</span>&#160;<span class="comment">    then false is returned, otherwise other tests are used</span></div>
<div class="line"><a name="l01291"></a><span class="lineno"> 1291</span>&#160;<span class="comment">    to determing continuity.</span></div>
<div class="line"><a name="l01292"></a><span class="lineno"> 1292</span>&#160;<span class="comment">  zero_curvature - [in] (ignored if &lt; 2^-110 = 7.7037197787136e-34)</span></div>
<div class="line"><a name="l01293"></a><span class="lineno"> 1293</span>&#160;<span class="comment">    If |K| &lt;= zero_curvature, then K is treated as zero.</span></div>
<div class="line"><a name="l01294"></a><span class="lineno"> 1294</span>&#160;<span class="comment">    When in doubt, use ON_ZERO_CURVATURE_TOLERANCE.</span></div>
<div class="line"><a name="l01295"></a><span class="lineno"> 1295</span>&#160;<span class="comment">  radius_tolerance - [in]</span></div>
<div class="line"><a name="l01296"></a><span class="lineno"> 1296</span>&#160;<span class="comment">    If radius_tolerance &gt;= 0.0 and the difference between the</span></div>
<div class="line"><a name="l01297"></a><span class="lineno"> 1297</span>&#160;<span class="comment">    radii of curvature is &gt;= radius_tolerance, then true </span></div>
<div class="line"><a name="l01298"></a><span class="lineno"> 1298</span>&#160;<span class="comment">    is returned.</span></div>
<div class="line"><a name="l01299"></a><span class="lineno"> 1299</span>&#160;<span class="comment">  relative_tolerance - [in]</span></div>
<div class="line"><a name="l01300"></a><span class="lineno"> 1300</span>&#160;<span class="comment">    If relative_tolerance &gt; 0 and</span></div>
<div class="line"><a name="l01301"></a><span class="lineno"> 1301</span>&#160;<span class="comment">    |(|Km| - |Kp|)|/max(|Km|,|Kp|) &gt; relative_tolerance,</span></div>
<div class="line"><a name="l01302"></a><span class="lineno"> 1302</span>&#160;<span class="comment">    then true is returned.  Note that if the curvatures are</span></div>
<div class="line"><a name="l01303"></a><span class="lineno"> 1303</span>&#160;<span class="comment">    nonzero and rm and rp are the radii of curvature, then</span></div>
<div class="line"><a name="l01304"></a><span class="lineno"> 1304</span>&#160;<span class="comment">    |(|Km| - |Kp|)|/max(|Km|,|Kp|) = |rm-rp|/max(rm,rp).</span></div>
<div class="line"><a name="l01305"></a><span class="lineno"> 1305</span>&#160;<span class="comment">    This means the relative_tolerance insures both the scalar</span></div>
<div class="line"><a name="l01306"></a><span class="lineno"> 1306</span>&#160;<span class="comment">    curvature and the radii of curvature agree to the specified</span></div>
<div class="line"><a name="l01307"></a><span class="lineno"> 1307</span>&#160;<span class="comment">    number of decimal places.</span></div>
<div class="line"><a name="l01308"></a><span class="lineno"> 1308</span>&#160;<span class="comment">    When in double use ON_RELATIVE_CURVATURE_TOLERANCE, which</span></div>
<div class="line"><a name="l01309"></a><span class="lineno"> 1309</span>&#160;<span class="comment">    is currently 0.05.</span></div>
<div class="line"><a name="l01310"></a><span class="lineno"> 1310</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01311"></a><span class="lineno"> 1311</span>&#160;<span class="comment">  False if the curvatures should be considered G2.</span></div>
<div class="line"><a name="l01312"></a><span class="lineno"> 1312</span>&#160;<span class="comment">  True if the curvatures are different enough that the curve should be</span></div>
<div class="line"><a name="l01313"></a><span class="lineno"> 1313</span>&#160;<span class="comment">  considered not G2.  </span></div>
<div class="line"><a name="l01314"></a><span class="lineno"> 1314</span>&#160;<span class="comment">  In addition to the tests described under the curvature_tolerance and </span></div>
<div class="line"><a name="l01315"></a><span class="lineno"> 1315</span>&#160;<span class="comment">  radius_tolerance checks, other hurestic tests are used.</span></div>
<div class="line"><a name="l01316"></a><span class="lineno"> 1316</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01317"></a><span class="lineno"> 1317</span>&#160;ON_DECL</div>
<div class="line"><a name="l01318"></a><span class="lineno"> 1318</span>&#160;<span class="keywordtype">bool</span> ON_IsCurvatureDiscontinuity( </div>
<div class="line"><a name="l01319"></a><span class="lineno"> 1319</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Km, </div>
<div class="line"><a name="l01320"></a><span class="lineno"> 1320</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Kp,</div>
<div class="line"><a name="l01321"></a><span class="lineno"> 1321</span>&#160;  <span class="keywordtype">double</span> cos_angle_tolerance,</div>
<div class="line"><a name="l01322"></a><span class="lineno"> 1322</span>&#160;  <span class="keywordtype">double</span> curvature_tolerance,</div>
<div class="line"><a name="l01323"></a><span class="lineno"> 1323</span>&#160;  <span class="keywordtype">double</span> zero_curvature,</div>
<div class="line"><a name="l01324"></a><span class="lineno"> 1324</span>&#160;  <span class="keywordtype">double</span> radius_tolerance,</div>
<div class="line"><a name="l01325"></a><span class="lineno"> 1325</span>&#160;  <span class="keywordtype">double</span> relative_tolerance</div>
<div class="line"><a name="l01326"></a><span class="lineno"> 1326</span>&#160;  );</div>
<div class="line"><a name="l01327"></a><span class="lineno"> 1327</span>&#160; </div>
<div class="line"><a name="l01328"></a><span class="lineno"> 1328</span>&#160;ON_DECL</div>
<div class="line"><a name="l01329"></a><span class="lineno"> 1329</span>&#160;<span class="keywordtype">bool</span> ON_IsCurvatureDiscontinuity( </div>
<div class="line"><a name="l01330"></a><span class="lineno"> 1330</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Km, </div>
<div class="line"><a name="l01331"></a><span class="lineno"> 1331</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Kp,</div>
<div class="line"><a name="l01332"></a><span class="lineno"> 1332</span>&#160;  <span class="keywordtype">double</span> cos_angle_tolerance,</div>
<div class="line"><a name="l01333"></a><span class="lineno"> 1333</span>&#160;  <span class="keywordtype">double</span> curvature_tolerance,</div>
<div class="line"><a name="l01334"></a><span class="lineno"> 1334</span>&#160;  <span class="keywordtype">double</span> zero_curvature,</div>
<div class="line"><a name="l01335"></a><span class="lineno"> 1335</span>&#160;  <span class="keywordtype">double</span> radius_tolerance</div>
<div class="line"><a name="l01336"></a><span class="lineno"> 1336</span>&#160;  );</div>
<div class="line"><a name="l01337"></a><span class="lineno"> 1337</span>&#160; </div>
<div class="line"><a name="l01338"></a><span class="lineno"> 1338</span>&#160; </div>
<div class="line"><a name="l01339"></a><span class="lineno"> 1339</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01340"></a><span class="lineno"> 1340</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01341"></a><span class="lineno"> 1341</span>&#160;<span class="comment">  This function is used to test curvature continuity</span></div>
<div class="line"><a name="l01342"></a><span class="lineno"> 1342</span>&#160;<span class="comment">  in IsContinuous and GetNextDiscontinuity functions</span></div>
<div class="line"><a name="l01343"></a><span class="lineno"> 1343</span>&#160;<span class="comment">  when the continuity parameter is ON::G2_continuous.</span></div>
<div class="line"><a name="l01344"></a><span class="lineno"> 1344</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01345"></a><span class="lineno"> 1345</span>&#160;<span class="comment">  Km - [in]</span></div>
<div class="line"><a name="l01346"></a><span class="lineno"> 1346</span>&#160;<span class="comment">    Curve&#39;s vector curvature evaluated from below</span></div>
<div class="line"><a name="l01347"></a><span class="lineno"> 1347</span>&#160;<span class="comment">  Kp - [in]</span></div>
<div class="line"><a name="l01348"></a><span class="lineno"> 1348</span>&#160;<span class="comment">    Curve&#39;s vector curvature evaluated from below</span></div>
<div class="line"><a name="l01349"></a><span class="lineno"> 1349</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01350"></a><span class="lineno"> 1350</span>&#160;<span class="comment">  True if the change from Km to Kp should be considered</span></div>
<div class="line"><a name="l01351"></a><span class="lineno"> 1351</span>&#160;<span class="comment">  G2 continuous.</span></div>
<div class="line"><a name="l01352"></a><span class="lineno"> 1352</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01353"></a><span class="lineno"> 1353</span>&#160;ON_DECL</div>
<div class="line"><a name="l01354"></a><span class="lineno"> 1354</span>&#160;<span class="keywordtype">bool</span> ON_IsG2CurvatureContinuous(</div>
<div class="line"><a name="l01355"></a><span class="lineno"> 1355</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Km, </div>
<div class="line"><a name="l01356"></a><span class="lineno"> 1356</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Kp,</div>
<div class="line"><a name="l01357"></a><span class="lineno"> 1357</span>&#160;  <span class="keywordtype">double</span> cos_angle_tolerance,</div>
<div class="line"><a name="l01358"></a><span class="lineno"> 1358</span>&#160;  <span class="keywordtype">double</span> curvature_tolerance</div>
<div class="line"><a name="l01359"></a><span class="lineno"> 1359</span>&#160;  );</div>
<div class="line"><a name="l01360"></a><span class="lineno"> 1360</span>&#160; </div>
<div class="line"><a name="l01361"></a><span class="lineno"> 1361</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01362"></a><span class="lineno"> 1362</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01363"></a><span class="lineno"> 1363</span>&#160;<span class="comment">  This function is used to test curvature continuity</span></div>
<div class="line"><a name="l01364"></a><span class="lineno"> 1364</span>&#160;<span class="comment">  in IsContinuous and GetNextDiscontinuity functions</span></div>
<div class="line"><a name="l01365"></a><span class="lineno"> 1365</span>&#160;<span class="comment">  when the continuity parameter is ON::Gsmooth_continuous.</span></div>
<div class="line"><a name="l01366"></a><span class="lineno"> 1366</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01367"></a><span class="lineno"> 1367</span>&#160;<span class="comment">  Km - [in]</span></div>
<div class="line"><a name="l01368"></a><span class="lineno"> 1368</span>&#160;<span class="comment">    Curve&#39;s vector curvature evaluated from below</span></div>
<div class="line"><a name="l01369"></a><span class="lineno"> 1369</span>&#160;<span class="comment">  Kp - [in]</span></div>
<div class="line"><a name="l01370"></a><span class="lineno"> 1370</span>&#160;<span class="comment">    Curve&#39;s vector curvature evaluated from below</span></div>
<div class="line"><a name="l01371"></a><span class="lineno"> 1371</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01372"></a><span class="lineno"> 1372</span>&#160;<span class="comment">  True if the change from Km to Kp should be considered</span></div>
<div class="line"><a name="l01373"></a><span class="lineno"> 1373</span>&#160;<span class="comment">  Gsmooth continuous.</span></div>
<div class="line"><a name="l01374"></a><span class="lineno"> 1374</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01375"></a><span class="lineno"> 1375</span>&#160;ON_DECL</div>
<div class="line"><a name="l01376"></a><span class="lineno"> 1376</span>&#160;<span class="keywordtype">bool</span> ON_IsGsmoothCurvatureContinuous(</div>
<div class="line"><a name="l01377"></a><span class="lineno"> 1377</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Km, </div>
<div class="line"><a name="l01378"></a><span class="lineno"> 1378</span>&#160;  <span class="keyword">const</span> <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> Kp,</div>
<div class="line"><a name="l01379"></a><span class="lineno"> 1379</span>&#160;  <span class="keywordtype">double</span> cos_angle_tolerance,</div>
<div class="line"><a name="l01380"></a><span class="lineno"> 1380</span>&#160;  <span class="keywordtype">double</span> curvature_tolerance</div>
<div class="line"><a name="l01381"></a><span class="lineno"> 1381</span>&#160;  );</div>
<div class="line"><a name="l01382"></a><span class="lineno"> 1382</span>&#160; </div>
<div class="line"><a name="l01383"></a><span class="lineno"> 1383</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01384"></a><span class="lineno"> 1384</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01385"></a><span class="lineno"> 1385</span>&#160;<span class="comment">  Test curve continuity from derivative values.</span></div>
<div class="line"><a name="l01386"></a><span class="lineno"> 1386</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01387"></a><span class="lineno"> 1387</span>&#160;<span class="comment">  c - [in] type of continuity to test for. Read ON::continuity</span></div>
<div class="line"><a name="l01388"></a><span class="lineno"> 1388</span>&#160;<span class="comment">           comments for details.</span></div>
<div class="line"><a name="l01389"></a><span class="lineno"> 1389</span>&#160;<span class="comment">  Pa - [in] point on curve A.</span></div>
<div class="line"><a name="l01390"></a><span class="lineno"> 1390</span>&#160;<span class="comment">  D1a - [in] first derviative of curve A.</span></div>
<div class="line"><a name="l01391"></a><span class="lineno"> 1391</span>&#160;<span class="comment">  D2a - [in] second derviative of curve A.</span></div>
<div class="line"><a name="l01392"></a><span class="lineno"> 1392</span>&#160;<span class="comment">  Pb - [in] point on curve B.</span></div>
<div class="line"><a name="l01393"></a><span class="lineno"> 1393</span>&#160;<span class="comment">  D1b - [in] first derviative of curve B.</span></div>
<div class="line"><a name="l01394"></a><span class="lineno"> 1394</span>&#160;<span class="comment">  D3b - [in] second derviative of curve B.</span></div>
<div class="line"><a name="l01395"></a><span class="lineno"> 1395</span>&#160;<span class="comment">  point_tolerance - [in] if the distance between two points is</span></div>
<div class="line"><a name="l01396"></a><span class="lineno"> 1396</span>&#160;<span class="comment">      greater than point_tolerance, then the curve is not C0.</span></div>
<div class="line"><a name="l01397"></a><span class="lineno"> 1397</span>&#160;<span class="comment">  d1_tolerance - [in] if the difference between two first derivatives is</span></div>
<div class="line"><a name="l01398"></a><span class="lineno"> 1398</span>&#160;<span class="comment">      greater than d1_tolerance, then the curve is not C1.</span></div>
<div class="line"><a name="l01399"></a><span class="lineno"> 1399</span>&#160;<span class="comment">  d2_tolerance - [in] if the difference between two second derivatives is</span></div>
<div class="line"><a name="l01400"></a><span class="lineno"> 1400</span>&#160;<span class="comment">      greater than d2_tolerance, then the curve is not C2.</span></div>
<div class="line"><a name="l01401"></a><span class="lineno"> 1401</span>&#160;<span class="comment">  cos_angle_tolerance - [in] default = cos(1 degree) Used only when</span></div>
<div class="line"><a name="l01402"></a><span class="lineno"> 1402</span>&#160;<span class="comment">      c is ON::G1_continuous or ON::G2_continuous.  If the cosine</span></div>
<div class="line"><a name="l01403"></a><span class="lineno"> 1403</span>&#160;<span class="comment">      of the angle between two tangent vectors </span></div>
<div class="line"><a name="l01404"></a><span class="lineno"> 1404</span>&#160;<span class="comment">      is &lt;= cos_angle_tolerance, then a G1 discontinuity is reported.</span></div>
<div class="line"><a name="l01405"></a><span class="lineno"> 1405</span>&#160;<span class="comment">  curvature_tolerance - [in] (default = ON_SQRT_EPSILON) Used only when</span></div>
<div class="line"><a name="l01406"></a><span class="lineno"> 1406</span>&#160;<span class="comment">      c is ON::G2_continuous.  If K0 and K1 are curvatures evaluated</span></div>
<div class="line"><a name="l01407"></a><span class="lineno"> 1407</span>&#160;<span class="comment">      from above and below and |K0 - K1| &gt; curvature_tolerance,</span></div>
<div class="line"><a name="l01408"></a><span class="lineno"> 1408</span>&#160;<span class="comment">      then a curvature discontinuity is reported.</span></div>
<div class="line"><a name="l01409"></a><span class="lineno"> 1409</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01410"></a><span class="lineno"> 1410</span>&#160;<span class="comment">  true if the curve has at least the c type continuity at </span></div>
<div class="line"><a name="l01411"></a><span class="lineno"> 1411</span>&#160;<span class="comment">  the parameter t.</span></div>
<div class="line"><a name="l01412"></a><span class="lineno"> 1412</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01413"></a><span class="lineno"> 1413</span>&#160;ON_DECL</div>
<div class="line"><a name="l01414"></a><span class="lineno"> 1414</span>&#160;ON_BOOL32 ON_IsContinuous(</div>
<div class="line"><a name="l01415"></a><span class="lineno"> 1415</span>&#160;  ON::continuity c,</div>
<div class="line"><a name="l01416"></a><span class="lineno"> 1416</span>&#160;  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a> Pa,</div>
<div class="line"><a name="l01417"></a><span class="lineno"> 1417</span>&#160;  <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> D1a,</div>
<div class="line"><a name="l01418"></a><span class="lineno"> 1418</span>&#160;  <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> D2a,</div>
<div class="line"><a name="l01419"></a><span class="lineno"> 1419</span>&#160;  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a> Pb,</div>
<div class="line"><a name="l01420"></a><span class="lineno"> 1420</span>&#160;  <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> D1b,</div>
<div class="line"><a name="l01421"></a><span class="lineno"> 1421</span>&#160;  <a class="code" href="class_o_n__3d_vector.html">ON_3dVector</a> D2b,</div>
<div class="line"><a name="l01422"></a><span class="lineno"> 1422</span>&#160;  <span class="keywordtype">double</span> point_tolerance=ON_ZERO_TOLERANCE,</div>
<div class="line"><a name="l01423"></a><span class="lineno"> 1423</span>&#160;  <span class="keywordtype">double</span> d1_tolerance=ON_ZERO_TOLERANCE,</div>
<div class="line"><a name="l01424"></a><span class="lineno"> 1424</span>&#160;  <span class="keywordtype">double</span> d2_tolerance=ON_ZERO_TOLERANCE,</div>
<div class="line"><a name="l01425"></a><span class="lineno"> 1425</span>&#160;  <span class="keywordtype">double</span> cos_angle_tolerance=ON_DEFAULT_ANGLE_TOLERANCE_COSINE,</div>
<div class="line"><a name="l01426"></a><span class="lineno"> 1426</span>&#160;  <span class="keywordtype">double</span> curvature_tolerance=ON_SQRT_EPSILON</div>
<div class="line"><a name="l01427"></a><span class="lineno"> 1427</span>&#160;  );</div>
<div class="line"><a name="l01428"></a><span class="lineno"> 1428</span>&#160; </div>
<div class="line"><a name="l01429"></a><span class="lineno"> 1429</span>&#160; </div>
<div class="line"><a name="l01430"></a><span class="lineno"> 1430</span>&#160;ON_DECL</div>
<div class="line"><a name="l01431"></a><span class="lineno"> 1431</span>&#160;<span class="keywordtype">bool</span> ON_TuneupEvaluationParameter( </div>
<div class="line"><a name="l01432"></a><span class="lineno"> 1432</span>&#160;   <span class="keywordtype">int</span> side,</div>
<div class="line"><a name="l01433"></a><span class="lineno"> 1433</span>&#160;   <span class="keywordtype">double</span> s0, <span class="keywordtype">double</span> s1, <span class="comment">// segment domain</span></div>
<div class="line"><a name="l01434"></a><span class="lineno"> 1434</span>&#160;   <span class="keywordtype">double</span> *s             <span class="comment">// segment parameter</span></div>
<div class="line"><a name="l01435"></a><span class="lineno"> 1435</span>&#160;   );</div>
<div class="line"><a name="l01436"></a><span class="lineno"> 1436</span>&#160; </div>
<div class="line"><a name="l01437"></a><span class="lineno"> 1437</span>&#160; </div>
<div class="line"><a name="l01438"></a><span class="lineno"> 1438</span>&#160;ON_DECL</div>
<div class="line"><a name="l01439"></a><span class="lineno"> 1439</span>&#160;<span class="keywordtype">int</span> ON_Compare2dex( <span class="keyword">const</span> <a class="code" href="structtag_o_n__2dex.html">ON_2dex</a>* a, <span class="keyword">const</span> <a class="code" href="structtag_o_n__2dex.html">ON_2dex</a>* b);</div>
<div class="line"><a name="l01440"></a><span class="lineno"> 1440</span>&#160; </div>
<div class="line"><a name="l01441"></a><span class="lineno"> 1441</span>&#160;ON_DECL</div>
<div class="line"><a name="l01442"></a><span class="lineno"> 1442</span>&#160;<span class="keywordtype">int</span> ON_Compare3dex( <span class="keyword">const</span> <a class="code" href="structtag_o_n__3dex.html">ON_3dex</a>* a, <span class="keyword">const</span> <a class="code" href="structtag_o_n__3dex.html">ON_3dex</a>* b);</div>
<div class="line"><a name="l01443"></a><span class="lineno"> 1443</span>&#160; </div>
<div class="line"><a name="l01444"></a><span class="lineno"> 1444</span>&#160;ON_DECL</div>
<div class="line"><a name="l01445"></a><span class="lineno"> 1445</span>&#160;<span class="keywordtype">int</span> ON_Compare4dex( <span class="keyword">const</span> <a class="code" href="structtag_o_n__4dex.html">ON_4dex</a>* a, <span class="keyword">const</span> <a class="code" href="structtag_o_n__4dex.html">ON_4dex</a>* b);</div>
<div class="line"><a name="l01446"></a><span class="lineno"> 1446</span>&#160; </div>
<div class="line"><a name="l01447"></a><span class="lineno"> 1447</span>&#160;ON_DECL</div>
<div class="line"><a name="l01448"></a><span class="lineno"> 1448</span>&#160;<span class="keyword">const</span> <a class="code" href="structtag_o_n__2dex.html">ON_2dex</a>* ON_BinarySearch2dexArray( </div>
<div class="line"><a name="l01449"></a><span class="lineno"> 1449</span>&#160;          <span class="keywordtype">int</span> key_i, </div>
<div class="line"><a name="l01450"></a><span class="lineno"> 1450</span>&#160;          <span class="keyword">const</span> <a class="code" href="structtag_o_n__2dex.html">ON_2dex</a>* base, </div>
<div class="line"><a name="l01451"></a><span class="lineno"> 1451</span>&#160;          <span class="keywordtype">size_t</span> nel</div>
<div class="line"><a name="l01452"></a><span class="lineno"> 1452</span>&#160;          );</div>
<div class="line"><a name="l01453"></a><span class="lineno"> 1453</span>&#160; </div>
<div class="line"><a name="l01454"></a><span class="lineno"> 1454</span>&#160;<span class="comment">// These simple intersectors are fast and detect transverse intersections.</span></div>
<div class="line"><a name="l01455"></a><span class="lineno"> 1455</span>&#160;<span class="comment">// If the intersection is not a simple transverse case, then they</span></div>
<div class="line"><a name="l01456"></a><span class="lineno"> 1456</span>&#160;<span class="comment">// return false and you will have to use one of the slower but fancier</span></div>
<div class="line"><a name="l01457"></a><span class="lineno"> 1457</span>&#160;<span class="comment">// models.</span></div>
<div class="line"><a name="l01458"></a><span class="lineno"> 1458</span>&#160; </div>
<div class="line"><a name="l01459"></a><span class="lineno"> 1459</span>&#160;<span class="comment">// returns closest points between the two infinite lines</span></div>
<div class="line"><a name="l01460"></a><span class="lineno"> 1460</span>&#160;ON_DECL</div>
<div class="line"><a name="l01461"></a><span class="lineno"> 1461</span>&#160;<span class="keywordtype">bool</span> ON_Intersect( </div>
<div class="line"><a name="l01462"></a><span class="lineno"> 1462</span>&#160;          <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;, </div>
<div class="line"><a name="l01463"></a><span class="lineno"> 1463</span>&#160;          <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;, </div>
<div class="line"><a name="l01464"></a><span class="lineno"> 1464</span>&#160;          <span class="keywordtype">double</span>*, <span class="comment">// parameter on first line</span></div>
<div class="line"><a name="l01465"></a><span class="lineno"> 1465</span>&#160;          <span class="keywordtype">double</span>*  <span class="comment">// parameter on second line</span></div>
<div class="line"><a name="l01466"></a><span class="lineno"> 1466</span>&#160;          );</div>
<div class="line"><a name="l01467"></a><span class="lineno"> 1467</span>&#160; </div>
<div class="line"><a name="l01468"></a><span class="lineno"> 1468</span>&#160;<span class="comment">// Returns false unless intersection is a single point</span></div>
<div class="line"><a name="l01469"></a><span class="lineno"> 1469</span>&#160;<span class="comment">// If returned parameter is &lt; 0 or &gt; 1, then the line</span></div>
<div class="line"><a name="l01470"></a><span class="lineno"> 1470</span>&#160;<span class="comment">// segment between line.m_point[0] and line.m_point[1]</span></div>
<div class="line"><a name="l01471"></a><span class="lineno"> 1471</span>&#160;<span class="comment">// does not intersect the plane</span></div>
<div class="line"><a name="l01472"></a><span class="lineno"> 1472</span>&#160;ON_DECL</div>
<div class="line"><a name="l01473"></a><span class="lineno"> 1473</span>&#160;<span class="keywordtype">bool</span> ON_Intersect( </div>
<div class="line"><a name="l01474"></a><span class="lineno"> 1474</span>&#160;          <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;, </div>
<div class="line"><a name="l01475"></a><span class="lineno"> 1475</span>&#160;          <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, </div>
<div class="line"><a name="l01476"></a><span class="lineno"> 1476</span>&#160;          <span class="keywordtype">double</span>* <span class="comment">// parameter on line</span></div>
<div class="line"><a name="l01477"></a><span class="lineno"> 1477</span>&#160;          );</div>
<div class="line"><a name="l01478"></a><span class="lineno"> 1478</span>&#160; </div>
<div class="line"><a name="l01479"></a><span class="lineno"> 1479</span>&#160;ON_DECL</div>
<div class="line"><a name="l01480"></a><span class="lineno"> 1480</span>&#160;<span class="keywordtype">bool</span> ON_Intersect( </div>
<div class="line"><a name="l01481"></a><span class="lineno"> 1481</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, </div>
<div class="line"><a name="l01482"></a><span class="lineno"> 1482</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, </div>
<div class="line"><a name="l01483"></a><span class="lineno"> 1483</span>&#160;        <a class="code" href="class_o_n___line.html">ON_Line</a>&amp; <span class="comment">// intersection line is returned here</span></div>
<div class="line"><a name="l01484"></a><span class="lineno"> 1484</span>&#160;        );</div>
<div class="line"><a name="l01485"></a><span class="lineno"> 1485</span>&#160; </div>
<div class="line"><a name="l01486"></a><span class="lineno"> 1486</span>&#160;ON_DECL</div>
<div class="line"><a name="l01487"></a><span class="lineno"> 1487</span>&#160;<span class="keywordtype">bool</span> ON_Intersect( </div>
<div class="line"><a name="l01488"></a><span class="lineno"> 1488</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, </div>
<div class="line"><a name="l01489"></a><span class="lineno"> 1489</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, </div>
<div class="line"><a name="l01490"></a><span class="lineno"> 1490</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;,</div>
<div class="line"><a name="l01491"></a><span class="lineno"> 1491</span>&#160;        <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; <span class="comment">// intersection point is returned here</span></div>
<div class="line"><a name="l01492"></a><span class="lineno"> 1492</span>&#160;        );</div>
<div class="line"><a name="l01493"></a><span class="lineno"> 1493</span>&#160; </div>
<div class="line"><a name="l01494"></a><span class="lineno"> 1494</span>&#160;<span class="comment">// returns 0 = no intersections, </span></div>
<div class="line"><a name="l01495"></a><span class="lineno"> 1495</span>&#160;<span class="comment">// 1 = intersection = single point, </span></div>
<div class="line"><a name="l01496"></a><span class="lineno"> 1496</span>&#160;<span class="comment">// 2 = intersection = circle</span></div>
<div class="line"><a name="l01497"></a><span class="lineno"> 1497</span>&#160;<span class="comment">// If 0 is returned, returned circle has radius=0</span></div>
<div class="line"><a name="l01498"></a><span class="lineno"> 1498</span>&#160;<span class="comment">// and center = point on sphere closest to plane.</span></div>
<div class="line"><a name="l01499"></a><span class="lineno"> 1499</span>&#160;<span class="comment">// If 1 is returned, intersection is a single</span></div>
<div class="line"><a name="l01500"></a><span class="lineno"> 1500</span>&#160;<span class="comment">// point and returned circle has radius=0</span></div>
<div class="line"><a name="l01501"></a><span class="lineno"> 1501</span>&#160;<span class="comment">// and center = intersection point on sphere.</span></div>
<div class="line"><a name="l01502"></a><span class="lineno"> 1502</span>&#160;ON_DECL</div>
<div class="line"><a name="l01503"></a><span class="lineno"> 1503</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01504"></a><span class="lineno"> 1504</span>&#160;                 <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;, <span class="keyword">const</span> <a class="code" href="class_o_n___sphere.html">ON_Sphere</a>&amp;, <a class="code" href="class_o_n___circle.html">ON_Circle</a>&amp;</div>
<div class="line"><a name="l01505"></a><span class="lineno"> 1505</span>&#160;                  );</div>
<div class="line"><a name="l01506"></a><span class="lineno"> 1506</span>&#160; </div>
<div class="line"><a name="l01507"></a><span class="lineno"> 1507</span>&#160;<span class="comment">// Intersects an infinte line and sphere and returns </span></div>
<div class="line"><a name="l01508"></a><span class="lineno"> 1508</span>&#160;<span class="comment">// 0 = no intersections, </span></div>
<div class="line"><a name="l01509"></a><span class="lineno"> 1509</span>&#160;<span class="comment">// 1 = one intersection, </span></div>
<div class="line"><a name="l01510"></a><span class="lineno"> 1510</span>&#160;<span class="comment">// 2 = 2 intersections</span></div>
<div class="line"><a name="l01511"></a><span class="lineno"> 1511</span>&#160;<span class="comment">// If 0 is returned, first point is point </span></div>
<div class="line"><a name="l01512"></a><span class="lineno"> 1512</span>&#160;<span class="comment">// on line closest to sphere and 2nd point is the point</span></div>
<div class="line"><a name="l01513"></a><span class="lineno"> 1513</span>&#160;<span class="comment">// on the sphere closest to the line.</span></div>
<div class="line"><a name="l01514"></a><span class="lineno"> 1514</span>&#160;<span class="comment">// If 1 is returned, first point is obtained by evaluating</span></div>
<div class="line"><a name="l01515"></a><span class="lineno"> 1515</span>&#160;<span class="comment">// the line and the second point is obtained by evaluating</span></div>
<div class="line"><a name="l01516"></a><span class="lineno"> 1516</span>&#160;<span class="comment">// the sphere.</span></div>
<div class="line"><a name="l01517"></a><span class="lineno"> 1517</span>&#160;ON_DECL</div>
<div class="line"><a name="l01518"></a><span class="lineno"> 1518</span>&#160;<span class="keywordtype">int</span> ON_Intersect(                  </div>
<div class="line"><a name="l01519"></a><span class="lineno"> 1519</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;, </div>
<div class="line"><a name="l01520"></a><span class="lineno"> 1520</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___sphere.html">ON_Sphere</a>&amp;,</div>
<div class="line"><a name="l01521"></a><span class="lineno"> 1521</span>&#160;        <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp;, </div>
<div class="line"><a name="l01522"></a><span class="lineno"> 1522</span>&#160;        <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; <span class="comment">// intersection point(s) returned here</span></div>
<div class="line"><a name="l01523"></a><span class="lineno"> 1523</span>&#160;        );</div>
<div class="line"><a name="l01524"></a><span class="lineno"> 1524</span>&#160; </div>
<div class="line"><a name="l01525"></a><span class="lineno"> 1525</span>&#160; </div>
<div class="line"><a name="l01526"></a><span class="lineno"> 1526</span>&#160;<span class="comment">// Intersects an infinte line and cylinder and returns </span></div>
<div class="line"><a name="l01527"></a><span class="lineno"> 1527</span>&#160;<span class="comment">// 0 = no intersections, </span></div>
<div class="line"><a name="l01528"></a><span class="lineno"> 1528</span>&#160;<span class="comment">// 1 = one intersection, </span></div>
<div class="line"><a name="l01529"></a><span class="lineno"> 1529</span>&#160;<span class="comment">// 2 = 2 intersections</span></div>
<div class="line"><a name="l01530"></a><span class="lineno"> 1530</span>&#160;<span class="comment">// 3 = line lies on cylinder</span></div>
<div class="line"><a name="l01531"></a><span class="lineno"> 1531</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l01532"></a><span class="lineno"> 1532</span>&#160;<span class="comment">// If 0 is returned, first point is point </span></div>
<div class="line"><a name="l01533"></a><span class="lineno"> 1533</span>&#160;<span class="comment">// on line closest to cylinder and 2nd point is the point</span></div>
<div class="line"><a name="l01534"></a><span class="lineno"> 1534</span>&#160;<span class="comment">// on the cylinder closest to the line.</span></div>
<div class="line"><a name="l01535"></a><span class="lineno"> 1535</span>&#160;<span class="comment">// If 1 is returned, first point is obtained by evaluating</span></div>
<div class="line"><a name="l01536"></a><span class="lineno"> 1536</span>&#160;<span class="comment">// the line and the second point is obtained by evaluating</span></div>
<div class="line"><a name="l01537"></a><span class="lineno"> 1537</span>&#160;<span class="comment">// the cylinder.</span></div>
<div class="line"><a name="l01538"></a><span class="lineno"> 1538</span>&#160;<span class="comment">//</span></div>
<div class="line"><a name="l01539"></a><span class="lineno"> 1539</span>&#160;<span class="comment">// The value of cylinder.IsFinite() determines if the</span></div>
<div class="line"><a name="l01540"></a><span class="lineno"> 1540</span>&#160;<span class="comment">// intersection is performed on the finite or infinite cylinder.</span></div>
<div class="line"><a name="l01541"></a><span class="lineno"> 1541</span>&#160;ON_DECL</div>
<div class="line"><a name="l01542"></a><span class="lineno"> 1542</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01543"></a><span class="lineno"> 1543</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;, <span class="comment">// [in]</span></div>
<div class="line"><a name="l01544"></a><span class="lineno"> 1544</span>&#160;      <span class="keyword">const</span> <a class="code" href="class_o_n___cylinder.html">ON_Cylinder</a>&amp;, <span class="comment">// [in]</span></div>
<div class="line"><a name="l01545"></a><span class="lineno"> 1545</span>&#160;      <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp;, <span class="comment">// [out] first intersection point</span></div>
<div class="line"><a name="l01546"></a><span class="lineno"> 1546</span>&#160;      <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; <span class="comment">// [out] second intersection point</span></div>
<div class="line"><a name="l01547"></a><span class="lineno"> 1547</span>&#160;      );</div>
<div class="line"><a name="l01548"></a><span class="lineno"> 1548</span>&#160; </div>
<div class="line"><a name="l01549"></a><span class="lineno"> 1549</span>&#160;<span class="comment">// Description:</span></div>
<div class="line"><a name="l01550"></a><span class="lineno"> 1550</span>&#160;<span class="comment">//   Intersect an infinte line and circle.</span></div>
<div class="line"><a name="l01551"></a><span class="lineno"> 1551</span>&#160;<span class="comment">// Parameters:</span></div>
<div class="line"><a name="l01552"></a><span class="lineno"> 1552</span>&#160;<span class="comment">//   line - [in]</span></div>
<div class="line"><a name="l01553"></a><span class="lineno"> 1553</span>&#160;<span class="comment">//   circle - [in]</span></div>
<div class="line"><a name="l01554"></a><span class="lineno"> 1554</span>&#160;<span class="comment">//   line_t0 - [out] line parameter of first intersection point</span></div>
<div class="line"><a name="l01555"></a><span class="lineno"> 1555</span>&#160;<span class="comment">//   circle_point0 - [out] first intersection point on circle</span></div>
<div class="line"><a name="l01556"></a><span class="lineno"> 1556</span>&#160;<span class="comment">//   line_t1 - [out] line parameter of second intersection point</span></div>
<div class="line"><a name="l01557"></a><span class="lineno"> 1557</span>&#160;<span class="comment">//   circle_point1 - [out] second intersection point on circle</span></div>
<div class="line"><a name="l01558"></a><span class="lineno"> 1558</span>&#160;<span class="comment">// Returns:</span></div>
<div class="line"><a name="l01559"></a><span class="lineno"> 1559</span>&#160;<span class="comment">//   0     No intersection</span></div>
<div class="line"><a name="l01560"></a><span class="lineno"> 1560</span>&#160;<span class="comment">//   1     One intersection at line.PointAt(*line_t0)</span></div>
<div class="line"><a name="l01561"></a><span class="lineno"> 1561</span>&#160;<span class="comment">//   2     Two intersections at line.PointAt(*line_t0)</span></div>
<div class="line"><a name="l01562"></a><span class="lineno"> 1562</span>&#160;<span class="comment">//         and line.PointAt(*line_t1).</span></div>
<div class="line"><a name="l01563"></a><span class="lineno"> 1563</span>&#160;ON_DECL</div>
<div class="line"><a name="l01564"></a><span class="lineno"> 1564</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01565"></a><span class="lineno"> 1565</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp; line, </div>
<div class="line"><a name="l01566"></a><span class="lineno"> 1566</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___circle.html">ON_Circle</a>&amp; circle,</div>
<div class="line"><a name="l01567"></a><span class="lineno"> 1567</span>&#160;                  <span class="keywordtype">double</span>* line_t0,</div>
<div class="line"><a name="l01568"></a><span class="lineno"> 1568</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; circle_point0,</div>
<div class="line"><a name="l01569"></a><span class="lineno"> 1569</span>&#160;                  <span class="keywordtype">double</span>* line_t1,</div>
<div class="line"><a name="l01570"></a><span class="lineno"> 1570</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; circle_point1</div>
<div class="line"><a name="l01571"></a><span class="lineno"> 1571</span>&#160;                  );</div>
<div class="line"><a name="l01572"></a><span class="lineno"> 1572</span>&#160; </div>
<div class="line"><a name="l01573"></a><span class="lineno"> 1573</span>&#160; </div>
<div class="line"><a name="l01574"></a><span class="lineno"> 1574</span>&#160; </div>
<div class="line"><a name="l01575"></a><span class="lineno"> 1575</span>&#160;<span class="comment">// Description:</span></div>
<div class="line"><a name="l01576"></a><span class="lineno"> 1576</span>&#160;<span class="comment">//   Intersect a infinte line and arc.</span></div>
<div class="line"><a name="l01577"></a><span class="lineno"> 1577</span>&#160;<span class="comment">// Parameters:</span></div>
<div class="line"><a name="l01578"></a><span class="lineno"> 1578</span>&#160;<span class="comment">//   line - [in]</span></div>
<div class="line"><a name="l01579"></a><span class="lineno"> 1579</span>&#160;<span class="comment">//   arc - [in]</span></div>
<div class="line"><a name="l01580"></a><span class="lineno"> 1580</span>&#160;<span class="comment">//   line_t0 - [out] line parameter of first intersection point</span></div>
<div class="line"><a name="l01581"></a><span class="lineno"> 1581</span>&#160;<span class="comment">//   arc_point0 - [out] first intersection point on arc</span></div>
<div class="line"><a name="l01582"></a><span class="lineno"> 1582</span>&#160;<span class="comment">//   line_t1 - [out] line parameter of second intersection point</span></div>
<div class="line"><a name="l01583"></a><span class="lineno"> 1583</span>&#160;<span class="comment">//   arc_point1 - [out] second intersection point on arc</span></div>
<div class="line"><a name="l01584"></a><span class="lineno"> 1584</span>&#160;<span class="comment">// Returns:</span></div>
<div class="line"><a name="l01585"></a><span class="lineno"> 1585</span>&#160;<span class="comment">//   0     No intersection</span></div>
<div class="line"><a name="l01586"></a><span class="lineno"> 1586</span>&#160;<span class="comment">//   1     One intersection at line.PointAt(*line_t0)</span></div>
<div class="line"><a name="l01587"></a><span class="lineno"> 1587</span>&#160;<span class="comment">//   2     Two intersections at line.PointAt(*line_t0)</span></div>
<div class="line"><a name="l01588"></a><span class="lineno"> 1588</span>&#160;<span class="comment">//         and line.PointAt(*line_t1).</span></div>
<div class="line"><a name="l01589"></a><span class="lineno"> 1589</span>&#160;ON_DECL</div>
<div class="line"><a name="l01590"></a><span class="lineno"> 1590</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01591"></a><span class="lineno"> 1591</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp; line, </div>
<div class="line"><a name="l01592"></a><span class="lineno"> 1592</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___arc.html">ON_Arc</a>&amp; arc,</div>
<div class="line"><a name="l01593"></a><span class="lineno"> 1593</span>&#160;                  <span class="keywordtype">double</span>* line_t0,</div>
<div class="line"><a name="l01594"></a><span class="lineno"> 1594</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; arc_point0,</div>
<div class="line"><a name="l01595"></a><span class="lineno"> 1595</span>&#160;                  <span class="keywordtype">double</span>* line_t1,</div>
<div class="line"><a name="l01596"></a><span class="lineno"> 1596</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; arc_point1</div>
<div class="line"><a name="l01597"></a><span class="lineno"> 1597</span>&#160;                  );</div>
<div class="line"><a name="l01598"></a><span class="lineno"> 1598</span>&#160; </div>
<div class="line"><a name="l01599"></a><span class="lineno"> 1599</span>&#160;<span class="comment">// Description:</span></div>
<div class="line"><a name="l01600"></a><span class="lineno"> 1600</span>&#160;<span class="comment">//   Intersect a plane and a circle.</span></div>
<div class="line"><a name="l01601"></a><span class="lineno"> 1601</span>&#160;<span class="comment">// Parameters:</span></div>
<div class="line"><a name="l01602"></a><span class="lineno"> 1602</span>&#160;<span class="comment">//   plane - [in]</span></div>
<div class="line"><a name="l01603"></a><span class="lineno"> 1603</span>&#160;<span class="comment">//   circle - [in]</span></div>
<div class="line"><a name="l01604"></a><span class="lineno"> 1604</span>&#160;<span class="comment">//   point0 - [out] first intersection point </span></div>
<div class="line"><a name="l01605"></a><span class="lineno"> 1605</span>&#160;<span class="comment">//   point1 - [out] second intersection point</span></div>
<div class="line"><a name="l01606"></a><span class="lineno"> 1606</span>&#160;<span class="comment">// Returns:</span></div>
<div class="line"><a name="l01607"></a><span class="lineno"> 1607</span>&#160;<span class="comment">//   0     No intersection</span></div>
<div class="line"><a name="l01608"></a><span class="lineno"> 1608</span>&#160;<span class="comment">//   1     One intersection at point0</span></div>
<div class="line"><a name="l01609"></a><span class="lineno"> 1609</span>&#160;<span class="comment">//   2     Two intersections at point0</span></div>
<div class="line"><a name="l01610"></a><span class="lineno"> 1610</span>&#160;<span class="comment">//         and point1.</span></div>
<div class="line"><a name="l01611"></a><span class="lineno"> 1611</span>&#160;<span class="comment">//   3       Circle lies on plane</span></div>
<div class="line"><a name="l01612"></a><span class="lineno"> 1612</span>&#160;ON_DECL</div>
<div class="line"><a name="l01613"></a><span class="lineno"> 1613</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01614"></a><span class="lineno"> 1614</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp; plane, </div>
<div class="line"><a name="l01615"></a><span class="lineno"> 1615</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___circle.html">ON_Circle</a>&amp; circle,</div>
<div class="line"><a name="l01616"></a><span class="lineno"> 1616</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; point0,</div>
<div class="line"><a name="l01617"></a><span class="lineno"> 1617</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; point1</div>
<div class="line"><a name="l01618"></a><span class="lineno"> 1618</span>&#160;                  );</div>
<div class="line"><a name="l01619"></a><span class="lineno"> 1619</span>&#160; </div>
<div class="line"><a name="l01620"></a><span class="lineno"> 1620</span>&#160;<span class="comment">// Description:</span></div>
<div class="line"><a name="l01621"></a><span class="lineno"> 1621</span>&#160;<span class="comment">//   Intersect a plane and an arc.</span></div>
<div class="line"><a name="l01622"></a><span class="lineno"> 1622</span>&#160;<span class="comment">// Parameters:</span></div>
<div class="line"><a name="l01623"></a><span class="lineno"> 1623</span>&#160;<span class="comment">//   plane - [in]</span></div>
<div class="line"><a name="l01624"></a><span class="lineno"> 1624</span>&#160;<span class="comment">//   arc - [in]</span></div>
<div class="line"><a name="l01625"></a><span class="lineno"> 1625</span>&#160;<span class="comment">//   point0 - [out] first intersection point </span></div>
<div class="line"><a name="l01626"></a><span class="lineno"> 1626</span>&#160;<span class="comment">//   point1 - [out] second intersection point</span></div>
<div class="line"><a name="l01627"></a><span class="lineno"> 1627</span>&#160;<span class="comment">// Returns:</span></div>
<div class="line"><a name="l01628"></a><span class="lineno"> 1628</span>&#160;<span class="comment">//   0     No intersection</span></div>
<div class="line"><a name="l01629"></a><span class="lineno"> 1629</span>&#160;<span class="comment">//   1     One intersection at point0</span></div>
<div class="line"><a name="l01630"></a><span class="lineno"> 1630</span>&#160;<span class="comment">//   2     Two intersections at point0</span></div>
<div class="line"><a name="l01631"></a><span class="lineno"> 1631</span>&#160;<span class="comment">//         and point1.</span></div>
<div class="line"><a name="l01632"></a><span class="lineno"> 1632</span>&#160;<span class="comment">//   3       Arc lies on plane</span></div>
<div class="line"><a name="l01633"></a><span class="lineno"> 1633</span>&#160;ON_DECL</div>
<div class="line"><a name="l01634"></a><span class="lineno"> 1634</span>&#160;<span class="keywordtype">int</span> ON_Intersect( </div>
<div class="line"><a name="l01635"></a><span class="lineno"> 1635</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp; plane, </div>
<div class="line"><a name="l01636"></a><span class="lineno"> 1636</span>&#160;                  <span class="keyword">const</span> <a class="code" href="class_o_n___arc.html">ON_Arc</a>&amp; arc,</div>
<div class="line"><a name="l01637"></a><span class="lineno"> 1637</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; point0,</div>
<div class="line"><a name="l01638"></a><span class="lineno"> 1638</span>&#160;                  <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>&amp; point1</div>
<div class="line"><a name="l01639"></a><span class="lineno"> 1639</span>&#160;                  );</div>
<div class="line"><a name="l01640"></a><span class="lineno"> 1640</span>&#160; </div>
<div class="line"><a name="l01641"></a><span class="lineno"> 1641</span>&#160; </div>
<div class="line"><a name="l01642"></a><span class="lineno"> 1642</span>&#160;<span class="comment">// returns 0 = no, 1 = yes, 2 = points are coincident and on line</span></div>
<div class="line"><a name="l01643"></a><span class="lineno"> 1643</span>&#160;ON_DECL</div>
<div class="line"><a name="l01644"></a><span class="lineno"> 1644</span>&#160;<span class="keywordtype">int</span> ON_ArePointsOnLine(</div>
<div class="line"><a name="l01645"></a><span class="lineno"> 1645</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// dimension of points</span></div>
<div class="line"><a name="l01646"></a><span class="lineno"> 1646</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// is_rat = true if homogeneous rational</span></div>
<div class="line"><a name="l01647"></a><span class="lineno"> 1647</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// count = number of points</span></div>
<div class="line"><a name="l01648"></a><span class="lineno"> 1648</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// stride ( &gt;= is_rat?(dim+1) :dim)</span></div>
<div class="line"><a name="l01649"></a><span class="lineno"> 1649</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// point array</span></div>
<div class="line"><a name="l01650"></a><span class="lineno"> 1650</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___bounding_box.html">ON_BoundingBox</a>&amp;, <span class="comment">// if needed, use ON_GetBoundingBox(dim,is_rat,count,stride,point)</span></div>
<div class="line"><a name="l01651"></a><span class="lineno"> 1651</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___line.html">ON_Line</a>&amp;,</div>
<div class="line"><a name="l01652"></a><span class="lineno"> 1652</span>&#160;        <span class="keywordtype">double</span>         <span class="comment">// tolerance (if 0.0, a tolerance based on bounding box size is used)</span></div>
<div class="line"><a name="l01653"></a><span class="lineno"> 1653</span>&#160;        );</div>
<div class="line"><a name="l01654"></a><span class="lineno"> 1654</span>&#160; </div>
<div class="line"><a name="l01655"></a><span class="lineno"> 1655</span>&#160;<span class="comment">// returns 0 = no, 1 = yes, 2 = points are coincident and on line</span></div>
<div class="line"><a name="l01656"></a><span class="lineno"> 1656</span>&#160;ON_DECL</div>
<div class="line"><a name="l01657"></a><span class="lineno"> 1657</span>&#160;<span class="keywordtype">int</span> ON_ArePointsOnPlane(</div>
<div class="line"><a name="l01658"></a><span class="lineno"> 1658</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// dimension of points</span></div>
<div class="line"><a name="l01659"></a><span class="lineno"> 1659</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// is_rat = true if homogeneous rational</span></div>
<div class="line"><a name="l01660"></a><span class="lineno"> 1660</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// count = number of points</span></div>
<div class="line"><a name="l01661"></a><span class="lineno"> 1661</span>&#160;        <span class="keywordtype">int</span>, <span class="comment">// stride ( &gt;= is_rat?(dim+1) :dim)</span></div>
<div class="line"><a name="l01662"></a><span class="lineno"> 1662</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// point array</span></div>
<div class="line"><a name="l01663"></a><span class="lineno"> 1663</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___bounding_box.html">ON_BoundingBox</a>&amp;, <span class="comment">// if needed, use ON_GetBoundingBox(dim,is_rat,count,stride,point)</span></div>
<div class="line"><a name="l01664"></a><span class="lineno"> 1664</span>&#160;        <span class="keyword">const</span> <a class="code" href="class_o_n___plane.html">ON_Plane</a>&amp;,</div>
<div class="line"><a name="l01665"></a><span class="lineno"> 1665</span>&#160;        <span class="keywordtype">double</span>         <span class="comment">// tolerance (if 0.0, a tolerance based on bounding box size is used)</span></div>
<div class="line"><a name="l01666"></a><span class="lineno"> 1666</span>&#160;        );</div>
<div class="line"><a name="l01667"></a><span class="lineno"> 1667</span>&#160; </div>
<div class="line"><a name="l01668"></a><span class="lineno"> 1668</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01669"></a><span class="lineno"> 1669</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01670"></a><span class="lineno"> 1670</span>&#160;<span class="comment">  Use the quotient rule to compute derivatives of a one parameter</span></div>
<div class="line"><a name="l01671"></a><span class="lineno"> 1671</span>&#160;<span class="comment">  rational function F(t) = X(t)/W(t), where W is a scalar</span></div>
<div class="line"><a name="l01672"></a><span class="lineno"> 1672</span>&#160;<span class="comment">  and F and X are vectors of dimension dim.</span></div>
<div class="line"><a name="l01673"></a><span class="lineno"> 1673</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01674"></a><span class="lineno"> 1674</span>&#160;<span class="comment">  dim - [in]</span></div>
<div class="line"><a name="l01675"></a><span class="lineno"> 1675</span>&#160;<span class="comment">  der_count - [in] number of derivative (&gt;=0)</span></div>
<div class="line"><a name="l01676"></a><span class="lineno"> 1676</span>&#160;<span class="comment">  v_stride - [in] (&gt;= dim+1)</span></div>
<div class="line"><a name="l01677"></a><span class="lineno"> 1677</span>&#160;<span class="comment">  v - [in/out]</span></div>
<div class="line"><a name="l01678"></a><span class="lineno"> 1678</span>&#160;<span class="comment">    v[] is an array of length (der_count+1)*v_stride.</span></div>
<div class="line"><a name="l01679"></a><span class="lineno"> 1679</span>&#160;<span class="comment">    The input v[] array contains  derivatives of the numerator and</span></div>
<div class="line"><a name="l01680"></a><span class="lineno"> 1680</span>&#160;<span class="comment">    denominator functions in the order (X, W), (Xt, Wt), (Xtt, Wtt), ...</span></div>
<div class="line"><a name="l01681"></a><span class="lineno"> 1681</span>&#160;<span class="comment">    In general, the (dim+1) coordinates of the d-th derivative </span></div>
<div class="line"><a name="l01682"></a><span class="lineno"> 1682</span>&#160;<span class="comment">    are in (v[n],...,v[n+dim]) where n = d*v_stride.</span></div>
<div class="line"><a name="l01683"></a><span class="lineno"> 1683</span>&#160;<span class="comment">    In the output v[] array the derivatives of X are replaced with</span></div>
<div class="line"><a name="l01684"></a><span class="lineno"> 1684</span>&#160;<span class="comment">    the derivatives of F and the derivatives of W are divided by</span></div>
<div class="line"><a name="l01685"></a><span class="lineno"> 1685</span>&#160;<span class="comment">    w = v[dim].</span></div>
<div class="line"><a name="l01686"></a><span class="lineno"> 1686</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01687"></a><span class="lineno"> 1687</span>&#160;<span class="comment">  True if input is valid; i.e., v[dim] != 0.</span></div>
<div class="line"><a name="l01688"></a><span class="lineno"> 1688</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l01689"></a><span class="lineno"> 1689</span>&#160;<span class="comment">  ON_EvaluateQuotientRule2</span></div>
<div class="line"><a name="l01690"></a><span class="lineno"> 1690</span>&#160;<span class="comment">  ON_EvaluateQuotientRule3</span></div>
<div class="line"><a name="l01691"></a><span class="lineno"> 1691</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01692"></a><span class="lineno"> 1692</span>&#160;ON_DECL</div>
<div class="line"><a name="l01693"></a><span class="lineno"> 1693</span>&#160;<span class="keywordtype">bool</span> ON_EvaluateQuotientRule( </div>
<div class="line"><a name="l01694"></a><span class="lineno"> 1694</span>&#160;          <span class="keywordtype">int</span> dim, </div>
<div class="line"><a name="l01695"></a><span class="lineno"> 1695</span>&#160;          <span class="keywordtype">int</span> der_count,</div>
<div class="line"><a name="l01696"></a><span class="lineno"> 1696</span>&#160;          <span class="keywordtype">int</span> v_stride, </div>
<div class="line"><a name="l01697"></a><span class="lineno"> 1697</span>&#160;          <span class="keywordtype">double</span> *v </div>
<div class="line"><a name="l01698"></a><span class="lineno"> 1698</span>&#160;          );</div>
<div class="line"><a name="l01699"></a><span class="lineno"> 1699</span>&#160; </div>
<div class="line"><a name="l01700"></a><span class="lineno"> 1700</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01701"></a><span class="lineno"> 1701</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01702"></a><span class="lineno"> 1702</span>&#160;<span class="comment">  Use the quotient rule to compute partial derivatives of a two parameter</span></div>
<div class="line"><a name="l01703"></a><span class="lineno"> 1703</span>&#160;<span class="comment">  rational function F(s,t) = X(s,t)/W(s,t), where W is a scalar</span></div>
<div class="line"><a name="l01704"></a><span class="lineno"> 1704</span>&#160;<span class="comment">  and F and X are vectors of dimension dim.</span></div>
<div class="line"><a name="l01705"></a><span class="lineno"> 1705</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01706"></a><span class="lineno"> 1706</span>&#160;<span class="comment">  dim - [in]</span></div>
<div class="line"><a name="l01707"></a><span class="lineno"> 1707</span>&#160;<span class="comment">  der_count - [in] number of derivative (&gt;=0)</span></div>
<div class="line"><a name="l01708"></a><span class="lineno"> 1708</span>&#160;<span class="comment">  v_stride - [in] (&gt;= dim+1)</span></div>
<div class="line"><a name="l01709"></a><span class="lineno"> 1709</span>&#160;<span class="comment">  v - [in/out]</span></div>
<div class="line"><a name="l01710"></a><span class="lineno"> 1710</span>&#160;<span class="comment">    v[] is an array of length (der_count+2)*(der_count+1)*v_stride.</span></div>
<div class="line"><a name="l01711"></a><span class="lineno"> 1711</span>&#160;<span class="comment">    The input array contains derivatives of the numerator and denominator</span></div>
<div class="line"><a name="l01712"></a><span class="lineno"> 1712</span>&#160;<span class="comment">        functions in the order X, W, Xs, Ws, Xt, Wt, Xss, Wss, Xst, Wst, Xtt, Wtt, ...</span></div>
<div class="line"><a name="l01713"></a><span class="lineno"> 1713</span>&#160;<span class="comment">    In general, the (i,j)-th derivatives are in the (dim+1) entries of v[]</span></div>
<div class="line"><a name="l01714"></a><span class="lineno"> 1714</span>&#160;<span class="comment">        v[k], ..., answer[k+dim], where k = ((i+j)*(i+j+1)/2 + j)*v_stride.</span></div>
<div class="line"><a name="l01715"></a><span class="lineno"> 1715</span>&#160;<span class="comment">    In the output v[] array the derivatives of X are replaced with</span></div>
<div class="line"><a name="l01716"></a><span class="lineno"> 1716</span>&#160;<span class="comment">    the derivatives of F and the derivatives of W are divided by</span></div>
<div class="line"><a name="l01717"></a><span class="lineno"> 1717</span>&#160;<span class="comment">    w = v[dim].</span></div>
<div class="line"><a name="l01718"></a><span class="lineno"> 1718</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01719"></a><span class="lineno"> 1719</span>&#160;<span class="comment">  True if input is valid; i.e., v[dim] != 0.</span></div>
<div class="line"><a name="l01720"></a><span class="lineno"> 1720</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l01721"></a><span class="lineno"> 1721</span>&#160;<span class="comment">  ON_EvaluateQuotientRule</span></div>
<div class="line"><a name="l01722"></a><span class="lineno"> 1722</span>&#160;<span class="comment">  ON_EvaluateQuotientRule3</span></div>
<div class="line"><a name="l01723"></a><span class="lineno"> 1723</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01724"></a><span class="lineno"> 1724</span>&#160;ON_DECL</div>
<div class="line"><a name="l01725"></a><span class="lineno"> 1725</span>&#160;<span class="keywordtype">bool</span> ON_EvaluateQuotientRule2( </div>
<div class="line"><a name="l01726"></a><span class="lineno"> 1726</span>&#160;          <span class="keywordtype">int</span> dim, </div>
<div class="line"><a name="l01727"></a><span class="lineno"> 1727</span>&#160;          <span class="keywordtype">int</span> der_count, </div>
<div class="line"><a name="l01728"></a><span class="lineno"> 1728</span>&#160;          <span class="keywordtype">int</span> v_stride, </div>
<div class="line"><a name="l01729"></a><span class="lineno"> 1729</span>&#160;          <span class="keywordtype">double</span> *v </div>
<div class="line"><a name="l01730"></a><span class="lineno"> 1730</span>&#160;          );</div>
<div class="line"><a name="l01731"></a><span class="lineno"> 1731</span>&#160; </div>
<div class="line"><a name="l01732"></a><span class="lineno"> 1732</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01733"></a><span class="lineno"> 1733</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01734"></a><span class="lineno"> 1734</span>&#160;<span class="comment">  Use the quotient rule to compute partial derivatives of a 3 parameter</span></div>
<div class="line"><a name="l01735"></a><span class="lineno"> 1735</span>&#160;<span class="comment">  rational function F(r,s,t) = X(r,s,t)/W(r,s,t), where W is a scalar</span></div>
<div class="line"><a name="l01736"></a><span class="lineno"> 1736</span>&#160;<span class="comment">  and F and X are vectors of dimension dim.</span></div>
<div class="line"><a name="l01737"></a><span class="lineno"> 1737</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01738"></a><span class="lineno"> 1738</span>&#160;<span class="comment">  dim - [in]</span></div>
<div class="line"><a name="l01739"></a><span class="lineno"> 1739</span>&#160;<span class="comment">  der_count - [in] number of derivative (&gt;=0)</span></div>
<div class="line"><a name="l01740"></a><span class="lineno"> 1740</span>&#160;<span class="comment">  v_stride - [in] (&gt;= dim+1)</span></div>
<div class="line"><a name="l01741"></a><span class="lineno"> 1741</span>&#160;<span class="comment">  v - [in/out]</span></div>
<div class="line"><a name="l01742"></a><span class="lineno"> 1742</span>&#160;<span class="comment">    v[] is an array of length </span></div>
<div class="line"><a name="l01743"></a><span class="lineno"> 1743</span>&#160;<span class="comment">    v_stride*(der_count+1)*(der_count+2)*(der_count+3)/6.</span></div>
<div class="line"><a name="l01744"></a><span class="lineno"> 1744</span>&#160;<span class="comment">    The input v[] array contains  derivatives of the numerator and</span></div>
<div class="line"><a name="l01745"></a><span class="lineno"> 1745</span>&#160;<span class="comment">    denominator functions in the order (X, W), (Xr, Wr), (Xs, Ws),</span></div>
<div class="line"><a name="l01746"></a><span class="lineno"> 1746</span>&#160;<span class="comment">    (Xt, Wt), (Xrr, Wrr), (Xrs, Wrs), (Xrt, Wrt), (Xss, Wss), </span></div>
<div class="line"><a name="l01747"></a><span class="lineno"> 1747</span>&#160;<span class="comment">    (Xst, Wst), (Xtt, Wtt), ...</span></div>
<div class="line"><a name="l01748"></a><span class="lineno"> 1748</span>&#160;<span class="comment">    In general, the (dim+1) coordinates of the derivative </span></div>
<div class="line"><a name="l01749"></a><span class="lineno"> 1749</span>&#160;<span class="comment">    (Dr^i Ds^j Dt^k, i+j+k=d) are at v[n], ..., v[n+dim] where </span></div>
<div class="line"><a name="l01750"></a><span class="lineno"> 1750</span>&#160;<span class="comment">    n = v_stride*( d*(d+1)*(d+2)/6  +  (d-i)*(d-i+1)/2  +  k ).</span></div>
<div class="line"><a name="l01751"></a><span class="lineno"> 1751</span>&#160;<span class="comment">    In the output v[] array the derivatives of X are replaced with</span></div>
<div class="line"><a name="l01752"></a><span class="lineno"> 1752</span>&#160;<span class="comment">    the derivatives of F and the derivatives of W are divided by</span></div>
<div class="line"><a name="l01753"></a><span class="lineno"> 1753</span>&#160;<span class="comment">    w = v[dim].</span></div>
<div class="line"><a name="l01754"></a><span class="lineno"> 1754</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01755"></a><span class="lineno"> 1755</span>&#160;<span class="comment">  True if input is valid; i.e., v[dim] != 0.</span></div>
<div class="line"><a name="l01756"></a><span class="lineno"> 1756</span>&#160;<span class="comment">See Also:</span></div>
<div class="line"><a name="l01757"></a><span class="lineno"> 1757</span>&#160;<span class="comment">  ON_EvaluateQuotientRule</span></div>
<div class="line"><a name="l01758"></a><span class="lineno"> 1758</span>&#160;<span class="comment">  ON_EvaluateQuotientRule2</span></div>
<div class="line"><a name="l01759"></a><span class="lineno"> 1759</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01760"></a><span class="lineno"> 1760</span>&#160;ON_DECL</div>
<div class="line"><a name="l01761"></a><span class="lineno"> 1761</span>&#160;<span class="keywordtype">bool</span> ON_EvaluateQuotientRule3( </div>
<div class="line"><a name="l01762"></a><span class="lineno"> 1762</span>&#160;          <span class="keywordtype">int</span> dim, </div>
<div class="line"><a name="l01763"></a><span class="lineno"> 1763</span>&#160;          <span class="keywordtype">int</span> der_count, </div>
<div class="line"><a name="l01764"></a><span class="lineno"> 1764</span>&#160;          <span class="keywordtype">int</span> v_stride,</div>
<div class="line"><a name="l01765"></a><span class="lineno"> 1765</span>&#160;          <span class="keywordtype">double</span> *v </div>
<div class="line"><a name="l01766"></a><span class="lineno"> 1766</span>&#160;          );</div>
<div class="line"><a name="l01767"></a><span class="lineno"> 1767</span>&#160; </div>
<div class="line"><a name="l01768"></a><span class="lineno"> 1768</span>&#160;ON_DECL</div>
<div class="line"><a name="l01769"></a><span class="lineno"> 1769</span>&#160;<span class="keywordtype">bool</span> ON_GetPolylineLength(</div>
<div class="line"><a name="l01770"></a><span class="lineno"> 1770</span>&#160;        <span class="keywordtype">int</span>,           <span class="comment">// dimension of points</span></div>
<div class="line"><a name="l01771"></a><span class="lineno"> 1771</span>&#160;        ON_BOOL32,          <span class="comment">// bIsRational true if points are homogeneous rational</span></div>
<div class="line"><a name="l01772"></a><span class="lineno"> 1772</span>&#160;        <span class="keywordtype">int</span>,           <span class="comment">// number of points</span></div>
<div class="line"><a name="l01773"></a><span class="lineno"> 1773</span>&#160;        <span class="keywordtype">int</span>,           <span class="comment">// stride between points</span></div>
<div class="line"><a name="l01774"></a><span class="lineno"> 1774</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>*, <span class="comment">// points</span></div>
<div class="line"><a name="l01775"></a><span class="lineno"> 1775</span>&#160;        <span class="keywordtype">double</span>*        <span class="comment">// length returned here</span></div>
<div class="line"><a name="l01776"></a><span class="lineno"> 1776</span>&#160;        );</div>
<div class="line"><a name="l01777"></a><span class="lineno"> 1777</span>&#160; </div>
<div class="line"><a name="l01778"></a><span class="lineno"> 1778</span>&#160; </div>
<div class="line"><a name="l01779"></a><span class="lineno"> 1779</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01780"></a><span class="lineno"> 1780</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01781"></a><span class="lineno"> 1781</span>&#160;<span class="comment">  Find the index of the point in the point_list that is closest to P.</span></div>
<div class="line"><a name="l01782"></a><span class="lineno"> 1782</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01783"></a><span class="lineno"> 1783</span>&#160;<span class="comment">  point_count - [in]</span></div>
<div class="line"><a name="l01784"></a><span class="lineno"> 1784</span>&#160;<span class="comment">  point_list - [in]</span></div>
<div class="line"><a name="l01785"></a><span class="lineno"> 1785</span>&#160;<span class="comment">  P - [in]</span></div>
<div class="line"><a name="l01786"></a><span class="lineno"> 1786</span>&#160;<span class="comment">  closest_point_index - [out]</span></div>
<div class="line"><a name="l01787"></a><span class="lineno"> 1787</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01788"></a><span class="lineno"> 1788</span>&#160;<span class="comment">  True if successful and *closest_point_index is set.</span></div>
<div class="line"><a name="l01789"></a><span class="lineno"> 1789</span>&#160;<span class="comment">  False if input is not valid, in which case *closest_point_index</span></div>
<div class="line"><a name="l01790"></a><span class="lineno"> 1790</span>&#160;<span class="comment">  is undefined.</span></div>
<div class="line"><a name="l01791"></a><span class="lineno"> 1791</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01792"></a><span class="lineno"> 1792</span>&#160;ON_DECL</div>
<div class="line"><a name="l01793"></a><span class="lineno"> 1793</span>&#160;<span class="keywordtype">bool</span> ON_GetClosestPointInPointList( </div>
<div class="line"><a name="l01794"></a><span class="lineno"> 1794</span>&#160;          <span class="keywordtype">int</span> point_count,</div>
<div class="line"><a name="l01795"></a><span class="lineno"> 1795</span>&#160;          <span class="keyword">const</span> <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a>* point_list,</div>
<div class="line"><a name="l01796"></a><span class="lineno"> 1796</span>&#160;          <a class="code" href="class_o_n__3d_point.html">ON_3dPoint</a> P,</div>
<div class="line"><a name="l01797"></a><span class="lineno"> 1797</span>&#160;          <span class="keywordtype">int</span>* closest_point_index</div>
<div class="line"><a name="l01798"></a><span class="lineno"> 1798</span>&#160;          );</div>
<div class="line"><a name="l01799"></a><span class="lineno"> 1799</span>&#160; </div>
<div class="line"><a name="l01800"></a><span class="lineno"> 1800</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01801"></a><span class="lineno"> 1801</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01802"></a><span class="lineno"> 1802</span>&#160;<span class="comment">  Test math library functions.</span></div>
<div class="line"><a name="l01803"></a><span class="lineno"> 1803</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01804"></a><span class="lineno"> 1804</span>&#160;<span class="comment">  function_index - [in]  Determines which math library function is called.</span></div>
<div class="line"><a name="l01805"></a><span class="lineno"> 1805</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l01806"></a><span class="lineno"> 1806</span>&#160;<span class="comment">           1:    z = x+y</span></div>
<div class="line"><a name="l01807"></a><span class="lineno"> 1807</span>&#160;<span class="comment">           2:    z = x-y</span></div>
<div class="line"><a name="l01808"></a><span class="lineno"> 1808</span>&#160;<span class="comment">           3:    z = x*y</span></div>
<div class="line"><a name="l01809"></a><span class="lineno"> 1809</span>&#160;<span class="comment">           4:    z = x/y</span></div>
<div class="line"><a name="l01810"></a><span class="lineno"> 1810</span>&#160;<span class="comment">           5:    z = fabs(x)</span></div>
<div class="line"><a name="l01811"></a><span class="lineno"> 1811</span>&#160;<span class="comment">           6:    z = exp(x)</span></div>
<div class="line"><a name="l01812"></a><span class="lineno"> 1812</span>&#160;<span class="comment">           7:    z = log(x)</span></div>
<div class="line"><a name="l01813"></a><span class="lineno"> 1813</span>&#160;<span class="comment">           8:    z = logb(x)</span></div>
<div class="line"><a name="l01814"></a><span class="lineno"> 1814</span>&#160;<span class="comment">           9:    z = log10(x)</span></div>
<div class="line"><a name="l01815"></a><span class="lineno"> 1815</span>&#160;<span class="comment">          10:    z = pow(x,y)</span></div>
<div class="line"><a name="l01816"></a><span class="lineno"> 1816</span>&#160;<span class="comment">          11:    z = sqrt(x)</span></div>
<div class="line"><a name="l01817"></a><span class="lineno"> 1817</span>&#160;<span class="comment">          12:    z = sin(x)</span></div>
<div class="line"><a name="l01818"></a><span class="lineno"> 1818</span>&#160;<span class="comment">          13:    z = cos(x)</span></div>
<div class="line"><a name="l01819"></a><span class="lineno"> 1819</span>&#160;<span class="comment">          14:    z = tan(x)</span></div>
<div class="line"><a name="l01820"></a><span class="lineno"> 1820</span>&#160;<span class="comment">          15:    z = sinh(x)</span></div>
<div class="line"><a name="l01821"></a><span class="lineno"> 1821</span>&#160;<span class="comment">          16:    z = cosh(x)</span></div>
<div class="line"><a name="l01822"></a><span class="lineno"> 1822</span>&#160;<span class="comment">          17:    z = tanh(x)</span></div>
<div class="line"><a name="l01823"></a><span class="lineno"> 1823</span>&#160;<span class="comment">          18:    z = asin(x)</span></div>
<div class="line"><a name="l01824"></a><span class="lineno"> 1824</span>&#160;<span class="comment">          19:    z = acos(x)</span></div>
<div class="line"><a name="l01825"></a><span class="lineno"> 1825</span>&#160;<span class="comment">          20:    z = atan(x)</span></div>
<div class="line"><a name="l01826"></a><span class="lineno"> 1826</span>&#160;<span class="comment">          21:    z = atan2(y,x)</span></div>
<div class="line"><a name="l01827"></a><span class="lineno"> 1827</span>&#160;<span class="comment">          22:    z = fmod(x,y)</span></div>
<div class="line"><a name="l01828"></a><span class="lineno"> 1828</span>&#160;<span class="comment">          23:    z = modf(x,&amp;y)</span></div>
<div class="line"><a name="l01829"></a><span class="lineno"> 1829</span>&#160;<span class="comment">          24:    z = frexp(x,&amp;y)</span></div>
<div class="line"><a name="l01830"></a><span class="lineno"> 1830</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l01831"></a><span class="lineno"> 1831</span>&#160;<span class="comment">  double x - [in]</span></div>
<div class="line"><a name="l01832"></a><span class="lineno"> 1832</span>&#160;<span class="comment">  double y - [in]</span></div>
<div class="line"><a name="l01833"></a><span class="lineno"> 1833</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01834"></a><span class="lineno"> 1834</span>&#160;<span class="comment">  Returns the &quot;z&quot; value listed in the function_index parameter</span></div>
<div class="line"><a name="l01835"></a><span class="lineno"> 1835</span>&#160;<span class="comment">  description.</span></div>
<div class="line"><a name="l01836"></a><span class="lineno"> 1836</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l01837"></a><span class="lineno"> 1837</span>&#160;<span class="comment">  This function is used to test the results of class floating</span></div>
<div class="line"><a name="l01838"></a><span class="lineno"> 1838</span>&#160;<span class="comment">  point functions.  It is primarily used to see what happens</span></div>
<div class="line"><a name="l01839"></a><span class="lineno"> 1839</span>&#160;<span class="comment">  when opennurbs is used as a DLL and illegal operations are</span></div>
<div class="line"><a name="l01840"></a><span class="lineno"> 1840</span>&#160;<span class="comment">  performed.</span></div>
<div class="line"><a name="l01841"></a><span class="lineno"> 1841</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01842"></a><span class="lineno"> 1842</span>&#160;ON_DECL</div>
<div class="line"><a name="l01843"></a><span class="lineno"> 1843</span>&#160;<span class="keywordtype">double</span> ON_TestMathFunction( </div>
<div class="line"><a name="l01844"></a><span class="lineno"> 1844</span>&#160;        <span class="keywordtype">int</span> function_index, </div>
<div class="line"><a name="l01845"></a><span class="lineno"> 1845</span>&#160;        <span class="keywordtype">double</span> x, </div>
<div class="line"><a name="l01846"></a><span class="lineno"> 1846</span>&#160;        <span class="keywordtype">double</span> y </div>
<div class="line"><a name="l01847"></a><span class="lineno"> 1847</span>&#160;        );</div>
<div class="line"><a name="l01848"></a><span class="lineno"> 1848</span>&#160; </div>
<div class="line"><a name="l01849"></a><span class="lineno"> 1849</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01850"></a><span class="lineno"> 1850</span>&#160;<span class="comment">// you are better off using ((b&lt;a)?a:b)</span></div>
<div class="line"><a name="l01851"></a><span class="lineno"> 1851</span>&#160;ON_DECL <span class="keywordtype">double</span> ON_Max(<span class="keywordtype">double</span> a, <span class="keywordtype">double</span> b);</div>
<div class="line"><a name="l01852"></a><span class="lineno"> 1852</span>&#160; </div>
<div class="line"><a name="l01853"></a><span class="lineno"> 1853</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01854"></a><span class="lineno"> 1854</span>&#160;<span class="comment">// you are better off using ((b&lt;a)?a:b)</span></div>
<div class="line"><a name="l01855"></a><span class="lineno"> 1855</span>&#160;ON_DECL <span class="keywordtype">float</span> ON_Max(<span class="keywordtype">float</span> a, <span class="keywordtype">float</span> b);</div>
<div class="line"><a name="l01856"></a><span class="lineno"> 1856</span>&#160; </div>
<div class="line"><a name="l01857"></a><span class="lineno"> 1857</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01858"></a><span class="lineno"> 1858</span>&#160;<span class="comment">// you are better off using ((b&lt;a)?a:b)</span></div>
<div class="line"><a name="l01859"></a><span class="lineno"> 1859</span>&#160;ON_DECL <span class="keywordtype">int</span> ON_Max(<span class="keywordtype">int</span> a, <span class="keywordtype">int</span> b);</div>
<div class="line"><a name="l01860"></a><span class="lineno"> 1860</span>&#160; </div>
<div class="line"><a name="l01861"></a><span class="lineno"> 1861</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01862"></a><span class="lineno"> 1862</span>&#160;<span class="comment">// you are better off using ((a&lt;b)?a:b)</span></div>
<div class="line"><a name="l01863"></a><span class="lineno"> 1863</span>&#160;ON_DECL <span class="keywordtype">double</span> ON_Min(<span class="keywordtype">double</span> a, <span class="keywordtype">double</span> b);</div>
<div class="line"><a name="l01864"></a><span class="lineno"> 1864</span>&#160; </div>
<div class="line"><a name="l01865"></a><span class="lineno"> 1865</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01866"></a><span class="lineno"> 1866</span>&#160;<span class="comment">// you are better off using ((a&lt;b)?a:b)</span></div>
<div class="line"><a name="l01867"></a><span class="lineno"> 1867</span>&#160;ON_DECL <span class="keywordtype">float</span> ON_Min(<span class="keywordtype">float</span> a, <span class="keywordtype">float</span> b);</div>
<div class="line"><a name="l01868"></a><span class="lineno"> 1868</span>&#160; </div>
<div class="line"><a name="l01869"></a><span class="lineno"> 1869</span>&#160;<span class="comment">// If performance is important, then</span></div>
<div class="line"><a name="l01870"></a><span class="lineno"> 1870</span>&#160;<span class="comment">// you are better off using ((a&lt;b)?a:b)</span></div>
<div class="line"><a name="l01871"></a><span class="lineno"> 1871</span>&#160;ON_DECL <span class="keywordtype">int</span> ON_Min(<span class="keywordtype">int</span> a, <span class="keywordtype">int</span> b);</div>
<div class="line"><a name="l01872"></a><span class="lineno"> 1872</span>&#160; </div>
<div class="line"><a name="l01873"></a><span class="lineno"> 1873</span>&#160;<span class="comment">// Do not call ON_Round() in any opennurbs code, tl code</span></div>
<div class="line"><a name="l01874"></a><span class="lineno"> 1874</span>&#160;<span class="comment">// or any other code that does critical calculations or</span></div>
<div class="line"><a name="l01875"></a><span class="lineno"> 1875</span>&#160;<span class="comment">// when there is any possibility that x is invalid or</span></div>
<div class="line"><a name="l01876"></a><span class="lineno"> 1876</span>&#160;<span class="comment">// fabs(x)&gt;2147483647. Use floor(x+0.5) instead.</span></div>
<div class="line"><a name="l01877"></a><span class="lineno"> 1877</span>&#160;ON_DECL <span class="keywordtype">int</span> ON_Round(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l01878"></a><span class="lineno"> 1878</span>&#160; </div>
<div class="line"><a name="l01879"></a><span class="lineno"> 1879</span>&#160; </div>
<div class="line"><a name="l01880"></a><span class="lineno"> 1880</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01881"></a><span class="lineno"> 1881</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01882"></a><span class="lineno"> 1882</span>&#160;<span class="comment">  Find the equation of the parabola, ellipse or hyperbola </span></div>
<div class="line"><a name="l01883"></a><span class="lineno"> 1883</span>&#160;<span class="comment">  (non-degenerate conic) that passes through six distinct points.</span></div>
<div class="line"><a name="l01884"></a><span class="lineno"> 1884</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01885"></a><span class="lineno"> 1885</span>&#160;<span class="comment">  stride - [in] (&gt;=2) </span></div>
<div class="line"><a name="l01886"></a><span class="lineno"> 1886</span>&#160;<span class="comment">    points array stride</span></div>
<div class="line"><a name="l01887"></a><span class="lineno"> 1887</span>&#160;<span class="comment">  points2d - [in] (&gt;=2) </span></div>
<div class="line"><a name="l01888"></a><span class="lineno"> 1888</span>&#160;<span class="comment">    i-th point is (points[i*stride],points[i*stride+1])</span></div>
<div class="line"><a name="l01889"></a><span class="lineno"> 1889</span>&#160;<span class="comment">  conic - [out]</span></div>
<div class="line"><a name="l01890"></a><span class="lineno"> 1890</span>&#160;<span class="comment">    Coefficients of the conic equation.</span></div>
<div class="line"><a name="l01891"></a><span class="lineno"> 1891</span>&#160;<span class="comment">    The points on the conic satisfy the equation</span></div>
<div class="line"><a name="l01892"></a><span class="lineno"> 1892</span>&#160;<span class="comment">      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 </span></div>
<div class="line"><a name="l01893"></a><span class="lineno"> 1893</span>&#160;<span class="comment">        + conic[3]*x + conic[4]*y + conic[5]</span></div>
<div class="line"><a name="l01894"></a><span class="lineno"> 1894</span>&#160;<span class="comment">  max_pivot - [out] (can be null)</span></div>
<div class="line"><a name="l01895"></a><span class="lineno"> 1895</span>&#160;<span class="comment">  min_pivot - [out] (can be null)</span></div>
<div class="line"><a name="l01896"></a><span class="lineno"> 1896</span>&#160;<span class="comment">  zero_pivot - [out] (can be null)</span></div>
<div class="line"><a name="l01897"></a><span class="lineno"> 1897</span>&#160;<span class="comment">    If there are some near duplicates in the input point set,</span></div>
<div class="line"><a name="l01898"></a><span class="lineno"> 1898</span>&#160;<span class="comment">    the calculation is not stable.  If you want to get an</span></div>
<div class="line"><a name="l01899"></a><span class="lineno"> 1899</span>&#160;<span class="comment">    estimate of the validity of the solution, then inspect</span></div>
<div class="line"><a name="l01900"></a><span class="lineno"> 1900</span>&#160;<span class="comment">    the returned values.  max_pivot should around 1, </span></div>
<div class="line"><a name="l01901"></a><span class="lineno"> 1901</span>&#160;<span class="comment">    min_pivot should be &gt; 1e-4 or so, and zero_pivot should</span></div>
<div class="line"><a name="l01902"></a><span class="lineno"> 1902</span>&#160;<span class="comment">    be &lt; 1e-10 or so.  If the returned pivots don&#39;t satisify</span></div>
<div class="line"><a name="l01903"></a><span class="lineno"> 1903</span>&#160;<span class="comment">    these condtions, then exercise caution when using the</span></div>
<div class="line"><a name="l01904"></a><span class="lineno"> 1904</span>&#160;<span class="comment">    returned solution.</span></div>
<div class="line"><a name="l01905"></a><span class="lineno"> 1905</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01906"></a><span class="lineno"> 1906</span>&#160;<span class="comment">  True if a there is an ellipse, parabola or hyperbola through the  </span></div>
<div class="line"><a name="l01907"></a><span class="lineno"> 1907</span>&#160;<span class="comment">  six points.</span></div>
<div class="line"><a name="l01908"></a><span class="lineno"> 1908</span>&#160;<span class="comment">  False if the input is invalid or the conic degenerate (the</span></div>
<div class="line"><a name="l01909"></a><span class="lineno"> 1909</span>&#160;<span class="comment">  points lie on one or two lines).</span></div>
<div class="line"><a name="l01910"></a><span class="lineno"> 1910</span>&#160;<span class="comment">  If false is returned, then conic[0]=...=conic[5] = 0 and</span></div>
<div class="line"><a name="l01911"></a><span class="lineno"> 1911</span>&#160;<span class="comment">  *min_pivot = *max_pivot = *zero_pivot = 0.</span></div>
<div class="line"><a name="l01912"></a><span class="lineno"> 1912</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01913"></a><span class="lineno"> 1913</span>&#160;ON_DECL <span class="keywordtype">bool</span> ON_GetConicEquationThrough6Points( </div>
<div class="line"><a name="l01914"></a><span class="lineno"> 1914</span>&#160;        <span class="keywordtype">int</span> stride, </div>
<div class="line"><a name="l01915"></a><span class="lineno"> 1915</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span>* points2d, </div>
<div class="line"><a name="l01916"></a><span class="lineno"> 1916</span>&#160;        <span class="keywordtype">double</span> conic[6],</div>
<div class="line"><a name="l01917"></a><span class="lineno"> 1917</span>&#160;        <span class="keywordtype">double</span>* max_pivot,</div>
<div class="line"><a name="l01918"></a><span class="lineno"> 1918</span>&#160;        <span class="keywordtype">double</span>* min_pivot,</div>
<div class="line"><a name="l01919"></a><span class="lineno"> 1919</span>&#160;        <span class="keywordtype">double</span>* zero_pivot</div>
<div class="line"><a name="l01920"></a><span class="lineno"> 1920</span>&#160;        );</div>
<div class="line"><a name="l01921"></a><span class="lineno"> 1921</span>&#160; </div>
<div class="line"><a name="l01922"></a><span class="lineno"> 1922</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01923"></a><span class="lineno"> 1923</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01924"></a><span class="lineno"> 1924</span>&#160;<span class="comment">  Test a conic equation to see if it defines and ellipse. If so,</span></div>
<div class="line"><a name="l01925"></a><span class="lineno"> 1925</span>&#160;<span class="comment">  return the center and axes of the ellipse.</span></div>
<div class="line"><a name="l01926"></a><span class="lineno"> 1926</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01927"></a><span class="lineno"> 1927</span>&#160;<span class="comment">  conic - [in]</span></div>
<div class="line"><a name="l01928"></a><span class="lineno"> 1928</span>&#160;<span class="comment">    Coefficients of the conic equation.</span></div>
<div class="line"><a name="l01929"></a><span class="lineno"> 1929</span>&#160;<span class="comment">    The points on the conic satisfy the equation</span></div>
<div class="line"><a name="l01930"></a><span class="lineno"> 1930</span>&#160;<span class="comment">      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 </span></div>
<div class="line"><a name="l01931"></a><span class="lineno"> 1931</span>&#160;<span class="comment">        + conic[3]*x + conic[4]*y + conic[5]</span></div>
<div class="line"><a name="l01932"></a><span class="lineno"> 1932</span>&#160;<span class="comment">  center - [out]</span></div>
<div class="line"><a name="l01933"></a><span class="lineno"> 1933</span>&#160;<span class="comment">  major_axis - [out]</span></div>
<div class="line"><a name="l01934"></a><span class="lineno"> 1934</span>&#160;<span class="comment">  minor_axis - [out]</span></div>
<div class="line"><a name="l01935"></a><span class="lineno"> 1935</span>&#160;<span class="comment">  major_radius - [out]</span></div>
<div class="line"><a name="l01936"></a><span class="lineno"> 1936</span>&#160;<span class="comment">  minor_radius - [out]</span></div>
<div class="line"><a name="l01937"></a><span class="lineno"> 1937</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01938"></a><span class="lineno"> 1938</span>&#160;<span class="comment">  True if the conic is an ellipse and the center and axes were found.</span></div>
<div class="line"><a name="l01939"></a><span class="lineno"> 1939</span>&#160;<span class="comment">  False if the conic is not an ellipse, in which case the input values</span></div>
<div class="line"><a name="l01940"></a><span class="lineno"> 1940</span>&#160;<span class="comment">  of center, major_axis, minor_axis, major_radius, and minor_radius</span></div>
<div class="line"><a name="l01941"></a><span class="lineno"> 1941</span>&#160;<span class="comment">  are not changed.</span></div>
<div class="line"><a name="l01942"></a><span class="lineno"> 1942</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01943"></a><span class="lineno"> 1943</span>&#160;ON_DECL <span class="keywordtype">bool</span> ON_IsConicEquationAnEllipse( </div>
<div class="line"><a name="l01944"></a><span class="lineno"> 1944</span>&#160;        <span class="keyword">const</span> <span class="keywordtype">double</span> conic[6], </div>
<div class="line"><a name="l01945"></a><span class="lineno"> 1945</span>&#160;        <a class="code" href="class_o_n__2d_point.html">ON_2dPoint</a>&amp; center, </div>
<div class="line"><a name="l01946"></a><span class="lineno"> 1946</span>&#160;        <a class="code" href="class_o_n__2d_vector.html">ON_2dVector</a>&amp; major_axis, </div>
<div class="line"><a name="l01947"></a><span class="lineno"> 1947</span>&#160;        <a class="code" href="class_o_n__2d_vector.html">ON_2dVector</a>&amp; minor_axis, </div>
<div class="line"><a name="l01948"></a><span class="lineno"> 1948</span>&#160;        <span class="keywordtype">double</span>* major_radius, </div>
<div class="line"><a name="l01949"></a><span class="lineno"> 1949</span>&#160;        <span class="keywordtype">double</span>* minor_radius</div>
<div class="line"><a name="l01950"></a><span class="lineno"> 1950</span>&#160;        );</div>
<div class="line"><a name="l01951"></a><span class="lineno"> 1951</span>&#160; </div>
<div class="line"><a name="l01952"></a><span class="lineno"> 1952</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l01953"></a><span class="lineno"> 1953</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l01954"></a><span class="lineno"> 1954</span>&#160;<span class="comment">  Get the conic equation of an ellipse.</span></div>
<div class="line"><a name="l01955"></a><span class="lineno"> 1955</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l01956"></a><span class="lineno"> 1956</span>&#160;<span class="comment">  a - [in] (a&gt;0)</span></div>
<div class="line"><a name="l01957"></a><span class="lineno"> 1957</span>&#160;<span class="comment">  b - [in] (b&gt;0)</span></div>
<div class="line"><a name="l01958"></a><span class="lineno"> 1958</span>&#160;<span class="comment">    a and b are the lengths of the axes. Either one</span></div>
<div class="line"><a name="l01959"></a><span class="lineno"> 1959</span>&#160;<span class="comment">    may be largest and they can be equal.</span></div>
<div class="line"><a name="l01960"></a><span class="lineno"> 1960</span>&#160;<span class="comment">  x0 - [in]</span></div>
<div class="line"><a name="l01961"></a><span class="lineno"> 1961</span>&#160;<span class="comment">  y0 - [in]</span></div>
<div class="line"><a name="l01962"></a><span class="lineno"> 1962</span>&#160;<span class="comment">    (x0,y0) is the enter of the ellipse.</span></div>
<div class="line"><a name="l01963"></a><span class="lineno"> 1963</span>&#160;<span class="comment">  alpha - [in] (angle in radians)</span></div>
<div class="line"><a name="l01964"></a><span class="lineno"> 1964</span>&#160;<span class="comment">    When alpha is 0, a corresponds to the x-axis and</span></div>
<div class="line"><a name="l01965"></a><span class="lineno"> 1965</span>&#160;<span class="comment">    b corresponds to the y axis.  If alpha is non-zero</span></div>
<div class="line"><a name="l01966"></a><span class="lineno"> 1966</span>&#160;<span class="comment">    it specifies the rotation of these axes.</span></div>
<div class="line"><a name="l01967"></a><span class="lineno"> 1967</span>&#160;<span class="comment">  conic - [out]</span></div>
<div class="line"><a name="l01968"></a><span class="lineno"> 1968</span>&#160;<span class="comment">    Coefficients of the conic equation.</span></div>
<div class="line"><a name="l01969"></a><span class="lineno"> 1969</span>&#160;<span class="comment">    The points on the conic satisfy the equation</span></div>
<div class="line"><a name="l01970"></a><span class="lineno"> 1970</span>&#160;<span class="comment">      0 = conic[0]*x^2 + conic[1]*xy + conic[2]*y^2 </span></div>
<div class="line"><a name="l01971"></a><span class="lineno"> 1971</span>&#160;<span class="comment">        + conic[3]*x + conic[4]*y + conic[5]</span></div>
<div class="line"><a name="l01972"></a><span class="lineno"> 1972</span>&#160;<span class="comment">  center - [out]</span></div>
<div class="line"><a name="l01973"></a><span class="lineno"> 1973</span>&#160;<span class="comment">  major_axis - [out]</span></div>
<div class="line"><a name="l01974"></a><span class="lineno"> 1974</span>&#160;<span class="comment">  minor_axis - [out]</span></div>
<div class="line"><a name="l01975"></a><span class="lineno"> 1975</span>&#160;<span class="comment">  major_radius - [out]</span></div>
<div class="line"><a name="l01976"></a><span class="lineno"> 1976</span>&#160;<span class="comment">  minor_radius - [out]</span></div>
<div class="line"><a name="l01977"></a><span class="lineno"> 1977</span>&#160;<span class="comment">Remarks:</span></div>
<div class="line"><a name="l01978"></a><span class="lineno"> 1978</span>&#160;<span class="comment">  Here is the way to evaluate a point on the ellipse:</span></div>
<div class="line"><a name="l01979"></a><span class="lineno"> 1979</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l01980"></a><span class="lineno"> 1980</span>&#160;<span class="comment">          </span></div>
<div class="line"><a name="l01981"></a><span class="lineno"> 1981</span>&#160;<span class="comment">          double t = ellipse paramter in radians;</span></div>
<div class="line"><a name="l01982"></a><span class="lineno"> 1982</span>&#160;<span class="comment">          double x = a*cos(t);</span></div>
<div class="line"><a name="l01983"></a><span class="lineno"> 1983</span>&#160;<span class="comment">          double y = b*sin(t);</span></div>
<div class="line"><a name="l01984"></a><span class="lineno"> 1984</span>&#160;<span class="comment">          ON_2dPoint ellipse_point;</span></div>
<div class="line"><a name="l01985"></a><span class="lineno"> 1985</span>&#160;<span class="comment">          ellipse_point.x = x0 + x*cos(alpha) + y*sin(alpha);</span></div>
<div class="line"><a name="l01986"></a><span class="lineno"> 1986</span>&#160;<span class="comment">          ellipse_point.y = y0 - x*sin(alpha) + y*cos(alpha);</span></div>
<div class="line"><a name="l01987"></a><span class="lineno"> 1987</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l01988"></a><span class="lineno"> 1988</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l01989"></a><span class="lineno"> 1989</span>&#160;<span class="comment">  True if the input is valid and conic[] was filled in.</span></div>
<div class="line"><a name="l01990"></a><span class="lineno"> 1990</span>&#160;<span class="comment">  Falis if the input is not valid.  In this case the values in conic[]</span></div>
<div class="line"><a name="l01991"></a><span class="lineno"> 1991</span>&#160;<span class="comment">  are not changed.</span></div>
<div class="line"><a name="l01992"></a><span class="lineno"> 1992</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l01993"></a><span class="lineno"> 1993</span>&#160;ON_DECL <span class="keywordtype">bool</span> ON_GetEllipseConicEquation( </div>
<div class="line"><a name="l01994"></a><span class="lineno"> 1994</span>&#160;      <span class="keywordtype">double</span> a, <span class="keywordtype">double</span> b, </div>
<div class="line"><a name="l01995"></a><span class="lineno"> 1995</span>&#160;      <span class="keywordtype">double</span> x0, <span class="keywordtype">double</span> y0, </div>
<div class="line"><a name="l01996"></a><span class="lineno"> 1996</span>&#160;      <span class="keywordtype">double</span> alpha,</div>
<div class="line"><a name="l01997"></a><span class="lineno"> 1997</span>&#160;      <span class="keywordtype">double</span> conic[6]</div>
<div class="line"><a name="l01998"></a><span class="lineno"> 1998</span>&#160;      );</div>
<div class="line"><a name="l01999"></a><span class="lineno"> 1999</span>&#160; </div>
<div class="line"><a name="l02000"></a><span class="lineno"> 2000</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l02001"></a><span class="lineno"> 2001</span>&#160;<span class="comment">Descripton:</span></div>
<div class="line"><a name="l02002"></a><span class="lineno"> 2002</span>&#160;<span class="comment">  Return the length of a 2d vector (x,y)</span></div>
<div class="line"><a name="l02003"></a><span class="lineno"> 2003</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l02004"></a><span class="lineno"> 2004</span>&#160;<span class="comment"> sqrt(x^2 + y^2) calculated in as precisely and safely as possible.</span></div>
<div class="line"><a name="l02005"></a><span class="lineno"> 2005</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l02006"></a><span class="lineno"> 2006</span>&#160;ON_DECL <span class="keywordtype">double</span> ON_Length2d( <span class="keywordtype">double</span> x, <span class="keywordtype">double</span> y );</div>
<div class="line"><a name="l02007"></a><span class="lineno"> 2007</span>&#160; </div>
<div class="line"><a name="l02008"></a><span class="lineno"> 2008</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l02009"></a><span class="lineno"> 2009</span>&#160;<span class="comment">Descripton:</span></div>
<div class="line"><a name="l02010"></a><span class="lineno"> 2010</span>&#160;<span class="comment">  Return the length of a 3d vector (x,y,z)</span></div>
<div class="line"><a name="l02011"></a><span class="lineno"> 2011</span>&#160;<span class="comment">Returns:</span></div>
<div class="line"><a name="l02012"></a><span class="lineno"> 2012</span>&#160;<span class="comment"> sqrt(x^2 + y^2 + z^2) calculated in as precisely and safely as possible.</span></div>
<div class="line"><a name="l02013"></a><span class="lineno"> 2013</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l02014"></a><span class="lineno"> 2014</span>&#160;ON_DECL <span class="keywordtype">double</span> ON_Length3d( <span class="keywordtype">double</span> x, <span class="keywordtype">double</span> y, <span class="keywordtype">double</span> z );</div>
<div class="line"><a name="l02015"></a><span class="lineno"> 2015</span>&#160; </div>
<div class="line"><a name="l02016"></a><span class="lineno"> 2016</span>&#160; </div>
<div class="line"><a name="l02017"></a><span class="lineno"> 2017</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l02018"></a><span class="lineno"> 2018</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l02019"></a><span class="lineno"> 2019</span>&#160;<span class="comment">  Convert a double x to the largest float f such that</span></div>
<div class="line"><a name="l02020"></a><span class="lineno"> 2020</span>&#160;<span class="comment">  the mathematical value of f is at most the value of x.</span></div>
<div class="line"><a name="l02021"></a><span class="lineno"> 2021</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l02022"></a><span class="lineno"> 2022</span>&#160;<span class="comment">  x - [in]</span></div>
<div class="line"><a name="l02023"></a><span class="lineno"> 2023</span>&#160;<span class="comment">Returns</span></div>
<div class="line"><a name="l02024"></a><span class="lineno"> 2024</span>&#160;<span class="comment">  The largest float f such that the mathematical value</span></div>
<div class="line"><a name="l02025"></a><span class="lineno"> 2025</span>&#160;<span class="comment">  of f is at most the value of x.</span></div>
<div class="line"><a name="l02026"></a><span class="lineno"> 2026</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l02027"></a><span class="lineno"> 2027</span>&#160;ON_DECL <span class="keywordtype">float</span> ON_FloatFloor(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l02028"></a><span class="lineno"> 2028</span>&#160; </div>
<div class="line"><a name="l02029"></a><span class="lineno"> 2029</span>&#160;<span class="comment">/*</span></div>
<div class="line"><a name="l02030"></a><span class="lineno"> 2030</span>&#160;<span class="comment">Description:</span></div>
<div class="line"><a name="l02031"></a><span class="lineno"> 2031</span>&#160;<span class="comment">  Convert a double x to the smallest float f such that</span></div>
<div class="line"><a name="l02032"></a><span class="lineno"> 2032</span>&#160;<span class="comment">  the mathematical value of f is at least the value of x.</span></div>
<div class="line"><a name="l02033"></a><span class="lineno"> 2033</span>&#160;<span class="comment">Parameters:</span></div>
<div class="line"><a name="l02034"></a><span class="lineno"> 2034</span>&#160;<span class="comment">  x - [in]</span></div>
<div class="line"><a name="l02035"></a><span class="lineno"> 2035</span>&#160;<span class="comment">Returns</span></div>
<div class="line"><a name="l02036"></a><span class="lineno"> 2036</span>&#160;<span class="comment">  The smallest float f such that the mathematical value</span></div>
<div class="line"><a name="l02037"></a><span class="lineno"> 2037</span>&#160;<span class="comment">  of f is at least the value of x.</span></div>
<div class="line"><a name="l02038"></a><span class="lineno"> 2038</span>&#160;<span class="comment">*/</span></div>
<div class="line"><a name="l02039"></a><span class="lineno"> 2039</span>&#160;ON_DECL <span class="keywordtype">float</span> ON_FloatCeil(<span class="keywordtype">double</span> x);</div>
<div class="line"><a name="l02040"></a><span class="lineno"> 2040</span>&#160; </div>
<div class="line"><a name="l02041"></a><span class="lineno"> 2041</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="ttc" id="aclass_o_n__2d_point_html"><div class="ttname"><a href="class_o_n__2d_point.html">ON_2dPoint</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:267</div></div>
<div class="ttc" id="aclass_o_n__2d_vector_html"><div class="ttname"><a href="class_o_n__2d_vector.html">ON_2dVector</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:675</div></div>
<div class="ttc" id="aclass_o_n__3d_point_html"><div class="ttname"><a href="class_o_n__3d_point.html">ON_3dPoint</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:419</div></div>
<div class="ttc" id="aclass_o_n__3d_vector_html"><div class="ttname"><a href="class_o_n__3d_vector.html">ON_3dVector</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:953</div></div>
<div class="ttc" id="aclass_o_n___arc_html"><div class="ttname"><a href="class_o_n___arc.html">ON_Arc</a></div><div class="ttdef"><b>Definition:</b> opennurbs_arc.h:35</div></div>
<div class="ttc" id="aclass_o_n___bounding_box_html"><div class="ttname"><a href="class_o_n___bounding_box.html">ON_BoundingBox</a></div><div class="ttdef"><b>Definition:</b> opennurbs_bounding_box.h:26</div></div>
<div class="ttc" id="aclass_o_n___circle_html"><div class="ttname"><a href="class_o_n___circle.html">ON_Circle</a></div><div class="ttdef"><b>Definition:</b> opennurbs_circle.h:34</div></div>
<div class="ttc" id="aclass_o_n___cylinder_html"><div class="ttname"><a href="class_o_n___cylinder.html">ON_Cylinder</a></div><div class="ttdef"><b>Definition:</b> opennurbs_cylinder.h:29</div></div>
<div class="ttc" id="aclass_o_n___evaluator_html"><div class="ttname"><a href="class_o_n___evaluator.html">ON_Evaluator</a></div><div class="ttdef"><b>Definition:</b> opennurbs_math.h:134</div></div>
<div class="ttc" id="aclass_o_n___interval_html"><div class="ttname"><a href="class_o_n___interval.html">ON_Interval</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:49</div></div>
<div class="ttc" id="aclass_o_n___line_html"><div class="ttname"><a href="class_o_n___line.html">ON_Line</a></div><div class="ttdef"><b>Definition:</b> opennurbs_line.h:21</div></div>
<div class="ttc" id="aclass_o_n___plane_equation_html"><div class="ttname"><a href="class_o_n___plane_equation.html">ON_PlaneEquation</a></div><div class="ttdef"><b>Definition:</b> opennurbs_point.h:1193</div></div>
<div class="ttc" id="aclass_o_n___plane_html"><div class="ttname"><a href="class_o_n___plane.html">ON_Plane</a></div><div class="ttdef"><b>Definition:</b> opennurbs_plane.h:21</div></div>
<div class="ttc" id="aclass_o_n___simple_array_html"><div class="ttname"><a href="class_o_n___simple_array.html">ON_SimpleArray&lt; ON_Interval &gt;</a></div></div>
<div class="ttc" id="aclass_o_n___sphere_html"><div class="ttname"><a href="class_o_n___sphere.html">ON_Sphere</a></div><div class="ttdef"><b>Definition:</b> opennurbs_sphere.h:23</div></div>
<div class="ttc" id="aclass_o_n___sum_html"><div class="ttname"><a href="class_o_n___sum.html">ON_Sum</a></div><div class="ttdef"><b>Definition:</b> opennurbs_math.h:31</div></div>
<div class="ttc" id="aclass_o_n___xform_html"><div class="ttname"><a href="class_o_n___xform.html">ON_Xform</a></div><div class="ttdef"><b>Definition:</b> opennurbs_xform.h:29</div></div>
<div class="ttc" id="astructtag_o_n__2dex_html"><div class="ttname"><a href="structtag_o_n__2dex.html">tagON_2dex</a></div><div class="ttdef"><b>Definition:</b> opennurbs_defines.h:314</div></div>
<div class="ttc" id="astructtag_o_n__3dex_html"><div class="ttname"><a href="structtag_o_n__3dex.html">tagON_3dex</a></div><div class="ttdef"><b>Definition:</b> opennurbs_defines.h:326</div></div>
<div class="ttc" id="astructtag_o_n__4dex_html"><div class="ttname"><a href="structtag_o_n__4dex.html">tagON_4dex</a></div><div class="ttdef"><b>Definition:</b> opennurbs_defines.h:340</div></div>
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